The X-Factor of Life

The X factor of life is an algebraic equation that analyzes our existence to give us victory in the end of times. In order to realize this equation we must first examine each of its parts.

PART A is titled Earth. Earth is the primary function in the equation. The numbers here are random and separate the different behavior patterns we all exhibit. They fall within two categories. Those of the negative nature encompass, rage, envy, jealousy, adultery, lasciviousness, hatred and anger. Those of the positive nature encompass long suffering, endurance, love, joy and peace. These parts are all random and are part of who we are as human beings. We are born into some and acquire others through life. Part A can be represented as follows:

PART A = Earth (Positive behavior + Negative behavior)

Where:

Positive = rage, envy, jealousy, adultery, lasciviousness, hatred, anger and many of this nature

Negative = love, joy, peace, long suffering, endurance and others of this nature

Part B is titled Doorway. This is the way, the truth and the life. They are a constant because they belong to our Lord and are him. Only by this doorway is anything prosperous possible for us in the end of time. The value of measurement for this is the flesh and blood, the spiritual intake in communion of our Lord and savior. Although we receive this as a blessing physically in church we must learn to accept it spiritually. This is represented as:

PART B = Doorway (Way + Truth + Life)

Where: way, truth and life are represented by flesh and blood as part of communion

When Part A and Part B are put together we get Part C which is spiritual prosperity. This is labeled Sheep. It is the combination of the element man, spirit existence and comforter since we have accepted Christ. The Lord's sheep hear his voice and come to him. They have the comforter within them to remind them ever so often of the Light of the Lord. This is the ignition within man to know and retain the knowledge of our Lord. Part C can be represented as follows:

PART C = Sheep {Comforter/spirit + man (Positive behavior >>Negative behavior)}

PART C = PART A + PART B

When these parts are combined correctly we worship God in truth and in spirit. It must be noted here that our spiritual existence cannot be measured by mathematical formula. These parts are simply expressed in this way to allow us to interpret the word of God in our lives.

Leslie Musoko
http://lesliemusoko.ning.com
http://lesliemusoko.blogspot.com

Saving Our Dropouts By Saving Math - Math Grades May Predict Who Survives High School

Research conducted in 2005 by Johns Hopkins University and the Philadelphia Education Fund revealed that as many as half of all Philadelphia high school dropouts showed signs predicting their early departure from school as early as the sixth grade. Four factors were essential in forecasting these AWOL students: low attendance, poor behavior, failing math, and failing English grades. Such research is indispensable in the fight to raise America's educational standards and to help struggling students conquer their academic nemeses.

It's difficult to extrapolate on precisely why so many leave school early, and why these occurrences are linked so strongly with math and English grades, specifically. Perhaps the current school system isn't flexible enough to accommodate different learning styles and conditions like ADHD for such difficult subjects - an argument alternative educational theorists have been raising for decades. Perhaps factors unrelated to the school environment, such as difficulties at home, are to blame. Maybe poor study techniques, and lack of assistance to correct them, are the culprits. Of course, no single explanation will do...but in my quest to find real and practical solutions, I decided to start with a factor everyone can control: study techniques. In particular, one of the hardest of them all - math study techniques.

Hey, who among us is, or was, the picture perfect student? Did we always clear a room to study, take breaks when we should have, or have what we needed on hand? How often did we get up to snack when the math homework became just a little too overwhelming? Did we always ask for the algebra help, the geometry help, or the calculus help we really needed? Math tutors could have saved us a lot of grief, but losing our pride just seemed too valuable to our adolescent hearts. Being guilty myself, I decided it was time to pull in a heavyweight for some advice.

Dr. Christine Benson, Associate Professor of Mathematics and Statistics at Northwest Missouri State University, is one of the most qualified individuals in the country to recommend effective math study techniques. Having earned an interdisciplinary doctorate in mathematics and education at the University of Missouri, Dr. Benson also received a MSEd, taught math at public schools for eight years, and has been teaching math method courses at NWMSU for the past twelve. Here are a few of her top recommendations for making the grade.

(1) Study in several short sessions of twenty to thirty minutes, and then take a break! There's only so much your poor mind can take at a time, and research proves that putting book to brain for shorter, intermittent periods, versus forcing yourself into longer cram sessions is far more effective. Plus (unlike most things that are good for you) you'll probably find this to be a lot easier.

(2) Study everyday. Math is a bit like learning a language - it takes consistent, sometimes tedious, irritating hours upon hours of work to get the job done. It also tends to pile up with cold indifference; at every lesson, you'll learn new concepts that count on your understanding of the ones from last time. Falling behind will just exponentialize your frustration, because you won't have the proper tools to understand new materials. Discipline pays off! Soon, you'll be able to understand concepts-the language of math, if you will-you never thought possible. You'll feel like the brilliant individual you really are. It just takes practice.

(3) Don't just memorize steps in an equation. "I know, I know," you say. "Try to understand it - that's what everyone tells me." But, like it or not, it's true. You'll remember the formulas much better if you can understand the bigger pictures behind them and are able to integrate new information into what you already know. Reasoning through questions that do not precisely fit the models you studied, but require you to use the broader concepts from them, will also be much easier.

(4) Ask questions! Don't be embarrassed to raise your hand and engage in a true discussion about all the whys of all these whats. You can't fully integrate the concepts you're trying to learn until you understand at least some of why they work and what they are meant to do.

(5) Include brain-empowering protein in your munchies. It's all right to snack while you study, but don't just reach for the carbs.

(6) If your eyes start to droop while pulling an all-nighter (or an all-dayer), wake yourself up with some physical activity. Get that circulation going! Take a walk, do some jumping jacks or push ups, go for a short jog - whatever it takes to change your setting and rouse your body.

(7) Duplicate the test room setting. If you'll be taking your exam in a quiet, controlled environment (which is most likely), then study in that environment. No music, no television, no loud noises or chatty friends coming in and out to distract you. Train your brain to work with the stimuli that will be there when you test.

(8) Get a good night's sleep and eat a protein-rich meal before the dreaded exam. You can't test your best if you don't feel your best.

At least something can be controlled. Failing at anything, or even not doing as well as you know you could, can be absolutely maddening. Dr. Benson has showed us there are things you can do, however. If you're a parent, enforce the rules. Set your child up for success by providing the proper environment. If you're a student, hey, you just got free expert advice - and you didn't even have to let anyone know you needed it.

Math Made Easy provides Math help for Algebra help, Geometry help, math homework help using math online tutorial services and math tutorial cd so you can watch your math scores soar.

Math Tutorial - Finding the Greatest Common Factor and Factoring Squares

Greatest Common Factor: of two numbers is the result of two numbers being factored into their smaller factors individually and the of all the numbers that are factors, the one that is greatest is the greatest common factor.

Symbol: ( ) means greatest common factor.

For instance, (6,8) means the greatest common factor of 6 and 8.

Example: (6, 8) is,

First, find all the factors of 6. The factors of 6 are all the numbers that when multiplied by another number give us 6. Those are, 1, 2, 3, 6. This is so because:

1*6=6 and 2*3=6

So you see that each number when multiplied by another number gives us 6.

Similarly,

Second, Find the factors of 8.

Those are, 1, 2, 4, 8.

So, now, when we look at the factors of 6 and 8

for 6: 1, 2, 3, 6

for 8: 1, 2, 4, 8

We see that 1, and 2, both appear in the factors of both six and 8:

Now, of the factors that appear in both numbers, that is 1 and 2, the greatest one is 2. *

We use greatest instead of greater because when we speak of bigger numbers, there might be more than just two numbers that are common factors.

Factoring A Difference Of Two Squares

Once students are familiar with the idea of factoring, there are shortcuts to the FOIL method of quadratic equation factoring. One of these is factoring a difference of two squares. A difference of two squares means that we have a monomial multiplied to another monomial to give us a quadratic function where one monomial is a conjugate pair of the other and the middle term in the quadratic form of an equation disappears.

A quadratic equation is represented by ax^2+bx+c, when the "bx" term disappears, we have a difference of two squares.

Example:

(x-3)*(x+3), here (x+3) is a conjugate pair of (x-3), and here is where we get a difference of two squares.

(x-3)(x+3), using the FOIL method, is:

(x)(x)+(x)(3)+(-3)(x)+(-3)(3)=

x^2+3x-3x-9

x^2-9.

Here, we see that since the middle term disappeared, we see that (x-3)(x+3) is simply X^2-(3)^2, the two squares being X and 3, which is why it is called the difference of two squares.

Of course, the same thing would result if we were doing (x+3)(x-3) since multiplication is commutative.

Now, to give more examples:

(x+4)(x-4)= (x^2-4^2)=(x^2-16)

(x+9)(x-9)=(x^2-9^2)=(x^2-81)

And so on, from here on, the patern is too clear and it would be too much repetition to go on.

Math Made Easy provides Math help for Algebra help, Geometry help, math homework help using math online tutorial services and math tutorial cd so you can watch your math scores soar.

Numerical Tools For College and University Students

Students attending college or university in a discipline heavy in the physical sciences, for example, science or engineering, frequently make use of several specific numerical routines. Five of the most popular numerical routines are examined below. These types of routines probably cover 90% of the routines a student will use during a typical undergraduate degree. In addition to their popularity among science and engineering programs, these numerical routines are also encountered in many other curriculum. For example, students in first year university who take an Algebra course to satisfy a breadth requirement might need a Simultaneous Equations Solver occasionally - while they are taking the course. Another student might need to apply a Linear Least Squares fit - once, for a specific assignment - when taking an Accounting class. If the students then continue on in their planned majors, say, Political Science or English, they do not use such tools again.

The five routines examined below are presented in response to the following hypothetical question: Which five numerical routines fill most - if not all - of the needs of undergraduate university students? The answer given below presents the most common types of numerical tasks and some of their applications. In addition, several good-quality free tools are named that offer solutions to these types of problems; they provide most of the functionality required by undergraduate students, allowing them to avoid - or at least delay - the expense of purchasing commercial software.

1) Root-finding

Root-finding covers the class of problem in which the zero(s) of an equation cannot be found explicitly.

Consider the Quadratic Equation:

a x^2 + b x + c = 0

a, b, and c are constants, and values of x that satisfy the equation, called the roots or zeros, must be found.

The Quadratic Equation is one example of the class of the problem of finding the roots of polynomial equations which is, in turn, part of the larger class of problem of root-finding. In fact, because the Quadratic Equation is so well-known (students are often introduced to the Quadratic Equation and its solution in Grade 10), root-finding is probably the best-known class of numerical routine.

The van der Waals Equation is another example of a polynomial equation for which roots are often sought:

pV^3 - n(RT + bp)V^2 + n^2 aV - n^3 ab = 0

In this case, values of V that satisfy the equation are sought, and the polynomial is a cubic (the highest power of V is 3). van der Waals Equation is often encountered in chemistry, thermodynamics, and gasdynamics applications.

Kepler's Equation of Elliptical Motion is another equation to which root-finding techniques are applied:

E - e sin(E) = M

In this example, the equation is not a polynomial, but it involves a transcendental function. e and M are known quantities, but there is no way to isolate E on one side of the equation and solve for it explicitly. Consequently, numerical techniques have to be employed. Rearranging the equation as follows turns the problem into one of finding the roots of the equation:

E - e sin(E) - M = 0

These examples are just three equations whose solution requires root-finding; many more equations arise whose solutions can be found only by employing root-finding techniques. Fortunately, the problem of root-finding is a well-developed field of mathematics and computer science. Almost all root-finding algorithms take an iterative approach to computing a solution to a desired degree of accuracy: first, an initial guess is made and checked, then a closer solution is estimated and checked, and this process is repeated until the user-specified level of accuracy is obtained. For example, a user might require four decimal places of accuracy in the solution, so the computer program would stop iterating for a solution once an approximation has been found to four decimal places.

2) Simultaneous Equations

This class of numerical task deals with solving N Equations in N Unknowns. For example, a situation may arise in which it can be mathematically described as a linear (the highest power of x present is 1) system of Three Equations in Three Unknowns:

a11 x1 + a12 x2 + a13 x3 = b1

a21 x1 + a22 x2 + a23 x3 = b2

a31 x1 + a32 x2 + a33 x3 = b3

The aii and bi values are known but the values of xi that satisfy this system of equations must be computed. This task could be accomplished with a pencil, paper, and hand calculator, but it would be tedious. And as systems get larger, the number of computations involved grows fast, introducing the risk of typos or other errors. A system of, say, 10 Equations in 10 Unknowns would keep a person busy for quite a while!

Fortunately, computer programs have been developed that can compute solutions to these systems quickly and accurately. They are usually put in matrix notation:

[A](x) = (b)

where [A] is a square matrix and (x) and (b) are column vectors.

These sorts of systems can arise from almost any field of study. In a course on Linear Algebra such systems will be faced all the time. These systems also arise in electric circuit analysis (i.e. - Mesh Current Analysis), industrial chemistry projects, structural analysis, economics studies, and more. In addition to solving the system for the x values, quantities of the [A] matrix itself are often computed to reveal informative properties (for example, its determinant, eigenvalues, and LU Decomposition).

3) Linear Least-Squares Data Fitting

Linear Least-Squares data fitting is often applied to describe data which includes errors. For example, a curve might be sought for data, but the data may be such that the expected curve does not satisfactorily pass through all the data points. For situations like this, a systematic method is required to produce an approximating function that describes the relationship defined by the data. The approximating function can then be used to interpolate data between the known data points (or to extrapolate outside the range of the known points). Linear least-squares data fitting is one tool available for such situations.

Applications for this class of numerical task arise in almost any field: economics, physics, politics, engineering, chemistry, environmental studies, and many more. For example, say a researcher has collected population data for a country over the past fifty years and would like to define an equation that effectively describes the population growth so that future growth can be extrapolated. Instead of simply looking at the data, and creating a "guesstimate" for an equation--a technique that would vary from one researcher to the next--a systematic and effective way of examining the data is offered by Linear Least- Squares Data Fitting; it offers a systematic approach for determining trends.

4) Interpolation

Interpolation is often used when drawing smooth curves through data, usually data that does not include errors, and provides a systematic technique for computing data values between the known data points (or outside the range of the known data points). For example, a researcher might have (x, y) data points for the following x-values: 1, 2, 3, 4, 5. However, the researcher might need a y-value that corresponds to an x-value of 2.5 or 6.4. The researcher would have to interpolate for the y-value at x = 2.5 (which is within the range of known data values) and extrapolate for the y-value at x = 6.4 (which is outside the range of known data values). Furthermore, the acquisition of the data may require sophisticated equipment that is hard to access, or the data may be very expensive to compute. In these sorts of situations, a systematic method of computing these interpolating data points is required.

Several algorithms exist for this purpose; one such algorithm is a Cubic Spline Interpolation. A Cubic Spline Interpolation creates a smooth curve through known data values by using piecewise third-degree polynomials that pass through all the data values. However, it should be noted that different versions of this algorithm exist, for example, a natural cubic spline interpolation has the second derivatives of the spline polynomial set to zero at the endpoints of the interpolation interval. This means that a graph of the spline outside the known data range is a straight line. Another version of the algorithm forces a "not-a-knot" condition: the second and second-last points are treated as interpolation points rather than knots (i.e. - the interpolating cubics on the first and second sub-intervals are identical, and so are the ones for the last and second last sub-intervals). Applications for spline interpolation include population data gathered over many years, cyclical sales information, and the contour of the shape of an automobile body.

5) Eigenvalues and Eigenvectors

lambda is an eigenvalue (a scalar) of the Matrix [A] if there is a non-zero vector (v) such that the following relationship is satisfied:

[A](v) = lambda (v)

Every vector (v) satisfying this equation is called an eigenvector of [A] belonging to the eigenvalue lambda.

Eigenproblems arise in almost all fields of science: structural analysis, computing the modes of vibration of a beam, aeroelasticity and flutter, system stability (structure, aircraft, satellites, etc.), heat transfer, biological systems, population growth, sociology, economics, and statistics. Eigenvalues and eigenvectors are also often used in conjunction with the solution of differential equations. Furthermore, the algorithm behind the Google search engine is also said to treat indexing as an eigenproblem.

Summary

Root-finding, solving Simultaneous Equations, Linear Least-Squares Data-fitting, Interpolation, and the computation of Eigenvalues and Eigenvectors are the most common types of problems faced by students in college and university. Not only are these types of numerical tasks faced by science and engineering students, they also show up throughout a variety of other programs. In addition, two more factors attest to the prevalence of these numerical problems: (i) routines for handling these types of tasks are almost always covered in texts and courses on numerical mathematics, and (ii) algorithms for these mathematical tasks are well-developed and source code for computer programs has been available for decades.

Considering their popularity, readily-available tools that provide solutions to these most common numerical tasks would appeal to a broad range of users. On one hand, some users might need a few routines for one-time or very infrequent use whereas, on the other hand, other users might use a program often, but only one specific routine. In either case, the purchase of a commercial software package is not justified and having free software available is a convenient alternative. In fact, these types of numerical math routines are widely available for free, in a variety of formats, offering a variety of capabilities. Several software packages have been developed for installation on a user's computer, for example, Octave and Scilab, to name two. Others are available as Java applets. And yet more are available as immediate-for-use Javascript web pages; for example, AKiTi.ca offers routines for solving many of these types of problems. The availability of these various numerical routines provides people more options when selecting a tool that best fits their unique needs, especially if these tools include solutions for the most common numerical tasks. The availability of good-quality software tools for working with the most common numerical tasks offers the greatest utility to the greatest number of people.

David Binner has a Bachelor's Degree in Engineering. Since graduating, he has taken up computer programming, with an emphasis on numerical programs, and web page design as hobbies. To contact David, please visit his web site, AKiTi.ca, and go to the "Contact" page.

Top 5 Mental Math Methods in the World

Today you can define mental math in various different ways. Some would say, memorizing times table and remembering the solutions can form the part of mental mathematics. Some would say ability to perform simple calculations in your head can be mental mathematics.

The web dictionary defines mental mathematics as "Computing an exact answer without using pencil and paper or other physical aids."

Today there are five methods available to learn and practice mental mathematics.

Let's begin with the first one called 'Learning by Heart' or better known as the rote memorizing method where your teachers ask you to mug up boring multiplication tables. It not only kills the interest of the child in mathematics but also makes sure that he develops hatred towards the subject for the rest of the years he studies it. This system gives its ardent devotee some degree of success initially as he is able to answer easy problems but then when the supposedly bigger application problems come the steam is almost over.

The second one gives you a good degree of success and I would highly recommend it to the younger lot out there. It hails from China and is popular by the name of The Abacus (also known as the Soroban in Japan). An abacus is a calculating tool, often constructed as a wooden frame with beads sliding on wires. With the use of this tool one can perform calculations relating to addition, subtraction, multiplication and division with ease. Gradually one practices with the tool in one's hand and later on when experienced he learns to do it without the tool. This tool is then fitted into the mind mentally and he can then add, subtract multiply and divide in seconds. This tool also enhances a child's concentration levels.

The main drawback of this system is that it focuses only on the 4 mathematical operations. Concepts beyond these operations such as Algebra, Square Roots, Cubes, Squares, Calculus, and Geometry etc cannot be solved using it at all. Also one needs a longer time to be able to fully get a grasp of the system hence you see courses in the abacus stretching to over 2 years which leads the child to boredom and then quitting from the course.

Another Chinese system mainly collected from the book The Nine Chapters on the Mathematical Art lays out an approach to mathematics that centers on finding the most general methods of solving problems. Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution, and an explanation of the procedure that led to the solution.

The methods explained in this system can hardly be termed mental and they lack speed to top it all. The Chinese were definitely the most advanced of the civilization thanks to the Yangtze and Yellow Rivers but if I were to choose out of the two methods given by this culture It would be the abacus.

If wars have a 99.99% downside, sometimes they can have an upside too for they give birth to stories of hope and creativity. The next mental math system was developed during the Second World War in the Nazi Concentration Camp by a Ukrainian Mathematician Jakow Trachtenberg to keep his mind occupied. What resulted is now known as the Trachtenberg Speed System of Mathematics and consists of Rapid Mental Methods of doing Mathematics.

The system consists of a number of readily memorized patterns that allow one to perform arithmetic computations very quickly. It has wider applications than the Abacus and apart from the four basic operation methods it covers Squares and Square Roots.

The method focuses mostly on Multiplication and it even gives patterns for multiplication by particular number say 5,6,7 and even 11 and 12. It then gives a general method for rapid multiplication and a special two finger method. After practicing the method myself I realized that the multiplication was a very applicable mental method but the other methods covered to solve division and square roots were not very friendly and were impossible to be done mentally. I was in search of a much better wholesome method where I could easily perform other operations also. Another drawback of this system was that it too like the abacus failed to have a wider scope i.e to encompass other fields like Algebra, Calculus, Trignometry, Cube Roots etc

A Recommendation by a friend of mine from America introduced me to what is known as the Kumon Math Method. It was founded by a Japanese educator Toru Kumon in 1950s and as of 2007 over 4 million children were studying under the Kumon Method in over 43 different countries.

Students do not work together as a class but progress through the curriculum at their own pace, moving on to the next level when they have achieved mastery of the previous level. This sometimes involves repeating the same set of worksheets until the student achieves a satisfactory score within a specified time limit. In North American Kumon Centers, the mathematics program starts with very basic skills, such as pattern recognition and counting, and progresses to increasingly challenging subjects, such as calculus, probability and statistics. The Kumon Method does not cover geometry as a separate topic but provides sufficient geometry practice to meet the prerequisites for trigonometry, which is covered within the Kumon math program.

I was much impressed with the glamour around Kumon but a glimpse of its curriculum deeply disappointed me. It is not mental at all. It does not offer any special methods to do mathematics and one does not improve one's speed by doing Kumon Math. There is a set curriculum of worksheets which one does till one achieves mastery in the subject. So say for example a sheet on Divison- one would continue to do division by the conventional method till he gets a satisfactory score and then he moves on to a higher level. This certainly doesn't make division any faster and the process is certainly not mental.

A deep thought on the reason of its tremendous popularity in America led me to conclude was the lack of a franchisee business model of the abacus and the Trachtenberg speed system in the 1950s. The franchisee model was essential in taking the course from country to country. This is where Toru Kumon thrived.

Dissapointed with other cultures in the world, my search made me look in my own Indian culture. What I found astonished and amazed me so much that I fell in love with the system and started coaching neighbourhood students in it.

This is easily the World's Fastest Mental Mathematics System called High Speed Vedic Mathematics. It has its roots in Ancient Indian Scriptures called the Vedas meaning 'the fountain head of knowledge'. With it not only you can add, subtract, multiply or divide which is the limiting factor of the abacus but you can also solve complex mathematics such as algebra, geometry, Calculus, and Trigonometry. Some of the most advanced, complex and arduous problems can be solved using the Vedic Maths method with extreme ease.

And all this with just 16 word formulas written in Sanskrit.

High Speed Vedic Mathematics was founded by Swami Sri Bharati Krishna Tirthaji Maharaja who was the Sankaracharya (Monk of the Highest Order) of Govardhan Matha in Puri between 1911 and 1918. They are called "Vedic" as because the sutras are contained in the Atharva Veda - a branch of mathematics and engineering in the Ancient Indian Scriptures.

High Speed Vedic Mathematics is far more systematic, simplified and unified than the conventional system. It is a mental tool for calculation that encourages the development and use of intuition and innovation, while giving the student a lot of flexibility, fun and satisfaction . For your child, it means giving them a competitive edge, a way to optimize their performance and gives them an edge in mathematics and logic that will help them to shine in the classroom and beyond.

Therefore it's direct and easy to implement in schools - a reason behind its enormous popularity among academicians and students. It complements the Mathematics curriculum conventionally taught in schools by acting as a powerful checking tool and goes to save precious time in examinations.

The Trachtenberg Method is often compared to Vedic Mathematics. Infact even some of the multiplication methods are strikingly similar. The Trachtenberg system comes the closest to the Vedic System in comparison and ease of the methods. But the ease and mental solvability of the other method especially division, square roots, cube roots, Algebraic Equations, Trigonometry, Calculus etc clearly gives the Vedic System an edge. Even NASA is said to be using some of this methods applications in the field of artificial intelligence.

There are just 16 Vedic Math sutras or word formulas which one needs to practice in order to be efficient in Vedic Math system. Sutras or Word Math Formulas such as the Vertically and Crosswise, All from Nine and Last from ten helps to solve complex problems with ease and also a single formula can be applied in two or more fields at the same time. The Vertically and Crosswise formula is one such gem by which one can multiply, find squares, solve simultaneous equations and find the determinant of a matrix all at the same time.

If either of these methods is learned at an early age, a student aged 14 can perform lightening fast calculations easily during his examinations and ace through them.

Vedic Mathematics is fast gaining popularity in this millennium. It is being considered as the only mental math system suited for a child as it helps to develop his numerical as well as mental abilities. The methods are new and practical and teach only Mental Rapid Mathematics.

The system does not focus on learning by repetition as in the Kumon Method. The system focuses on improving intelligence by teaching fundamentals and alternate methods. The purpose is not limited to improving performance in the school or tests, but on providing a broader outlook resulting in improved mathematical intelligence and mental sharpness.

To know more about the Vedic Mathematics Sutras - The World's Fastest Mental Math System you can visit http://www.vedicmathsindia.org

This Article is by Gaurav Tekriwal,, The President of the Vedic Maths Forum India who has been conducting High Speed Vedic Math Workshops for the last five years and has trained over seven thousand students across the world in the field. He is the author of the best selling DVD on the subject which contains over 10 hours on the subject. He is an expert in the field and revolutionizes the way children learn math.

This Article is by Gaurav Tekriwal, The President of the Vedic Maths Forum India who has been conducting High Speed Vedic Math Workshops for the last five years and has trained over seven thousand students across the world in the field. The author can be reached for consultancy on speed Vedic mathematics at gtekriwal@gmail.com

Math Tutoring for the Real World

If you're the parent of a student who is having a difficult time with math, you're familiar with the complaint that equations seem "pointless." Students want to know how solving equations with variables will help them in the real world. Students aren't sure that real people in the real world use math to solve real problems in their daily lives.

Students need to understand that logic used to solve math equations is a skill that will benefit them throughout life. Even in non-math situations, it is often necessary to isolate the unknown factor in a scenario to understand logically how to go about solving a certain problem. Math methodology lays the foundation for good problem-solving skills. In fact, many potential employers will require applicants to take a general math quiz before being hired. This realism makes the subject matter more enjoyable and easy to learn.

Many students who are struggling in their math work need additional tutorial assistance. In fact, hundreds of thousands of children having difficulty with a subject in school are currently being tutored in the United States for a variety of reasons:

· Many students didn't master basic skills which need to be re-taught to them

· Some have a learning disability which poses challenges to the mastery of math and slows down progress in school

· Others have weak organizational skills which result in difficulty with keeping on schedule with studying and completing assignments

· Some students have medical, social, emotional, behavioral and/or family problems which result in their struggling to keep up with their peers

· And still others simply desire to get ahead

Very often, all it takes to improve a student's low math grades is the right approach. Once that happens, it's as if a light comes on: suddenly everything falls into place! Even students who have been performing very poorly in math can finally experience the joy of "getting it."

Where can parents and students go for good quality math tutoring? Many parents find local tutors but rates can be as high as $175 per hour and not always effective.

An October 16th, 2007 Tutoring Report appeared on the NBC Today Show which described the dilemma many parents face in providing affordable quality tutoring.

As part of its overview of available services, the NBC Today Show explained that new internet technology is available to provide effective online tutoring services which enable students to get high quality one-on-one tutoring in the convenience of their own home at affordable prices.

With some online tutoring services, students can receive supplemental materials such as DVDs featuring complete review by expert teachers of the subject they are studying in school and unlimited practice exercises. In addition, online services may provide students access to new white board technology which enables them to watch the tutor's lesson and talk to them as if they were face to face with the tutors.

The NBC Today Show interviewed the Foley family in Peekskill, New York where Mrs. Foley acknowledged that she could not afford the costs of chain tutoring company services and therefore opted for an online tutoring program which was affordable and also gave her a full 30 day guarantee.

The two Foley daughters who used the program found that their math grades had a dramatic increase after using the online tutoring service and the DVD math reviews. In the interview, one of the Foley teenage girls said that she found the tutoring service easy to use and a really effective service.

Although no one wants their child to struggle, the good news is that solutions are available for math help. Parents should realize that the knowledge and sense of achievement that a tutoring program can bring to the student will pay big dividends for years to come. Not only will it help earn a better grade, but it will also lay a critical foundation for future success, including help with college admissions exams.

Math Made Easy provides Math help for Algebra help, Geometry help, math homework help using math online tutorial services and math tutorial cd so you can watch your math scores soar. For more information, please visit http://www.mathmadeeasy.com

Math Made Easy provides Math help for Algebra help, Geometry help, math homework help using math online tutorial services and math tutorial cd so you can watch your math scores soar.

What Makes Algebra So Difficult For Kids To Understand And Challenging for Teachers To Teach?

At first glance this seems to be another Chicken or the Egg situation. Are kids not understanding because Algebra is a subject that is difficult to teach or is Algebra difficult to teach because kids have so much trouble understanding it? In reality, there is some truth to both of these issues; and, theoretically, solving one will solve the other as well. So, exactly what is it that makes Algebra so unique?

Algebra is like a giant question mark in the brain of every freshman who walks into the classroom. These 14- and 15-year old students enter the classroom having absolutely no expectations of what they are going to be learning; and teaching Algebra successfully is one of the greatest educational challenges existing today.

Students in elementary school know just exactly what math was covered in each grade and what is coming next year. Addition, subtraction, multiplication,... They know. In high school, at the end of Geometry, Trigonometry, and Calculus, students can explain to you what the course was about. But Algebra is a different kind of animal. Too often Algebra teachers assume their students know what Algebra is, so Day One of school is Section One of Chapter One in the textbook and off they go on their unknown journey. Sadly, many students are as clueless at the end of the school year as they where at the beginning as to what they have been studying. Some students can tell you they solved(?) equations, they factored something, and they graphed things. Some students can actually be good at Algebra skills, but still have no idea why they were doing any of it. That's very sad.

The numerical skills required in Algebra (the HOW) are really pretty basic. It is the understanding of the WHY and WHEN that students don't get. But is this a student issue or a teacher issue?

Students issues to consider:

(1) knowledge of multiplication facts is the #1 indicator of success in Algebra, yet many students enter Algebra with weak multiplication skills,

(2) most students are lacking the ingrained sense of "I am smart enough" that they possessed when learning language,

(3) many students have lost the persistence they demonstrated when learning to walk, talk, and read,

(4) most students lack a pre-school math foundation similar to what parents provide for language skills,

(5) unlike all previous math courses in which only 25% of the material is new (never seen before), the amount of new material being covered in Algebra is approximately 75% of the course which seems to be too much for them to absorb,

(6) the pace required to cover so much new material seems too fast for students to absorb, and

(7) many Algebra students see no practical application to their lives, so they view it as unnecessary to learn. Have I missed any student issues? Probably, but you get the point.

Teacher issues to consider:

(1) the assumption that students already know what Algebra is is incorrect,

(2) teachers sometimes don't recognize that the problem is weak basic skills until the damage is done,

(3) the large amount of new material to be covered does not allow for proper processing but teachers do not have a choice about removing some of the subject matter,

(4) some teachers are weak at task analysis, (5) a few teachers have trouble explaining a topic several ways to deal with the different ways students learn, and

(6) No Child Left Behind has caused immeasurable harm to mathematics education and the learning environment. Again, you get my point even If I missed something.

In spite of all the issues I just listed, it should be noted that this "problem" has existed literally forever. The failure rate was 50% when I started teaching in 1972 and it still is. Many attempts have been made over the years to solve these issues. Nothing has been successful. So the answer to the initial question is: we don't know. If we knew, the issues would get solved.

Shirley Slick, "The Slick Tips Lady," is a retired high school math teacher and tutor with degrees in Mathematics and Psychology and additional training in brain-based learning/teaching. Her goals: (1) to help parents help their children with math, (2) to help eliminate the horrendous Algebra failure rate, and (3) to inform the general public about problematic issues related to the field of education. For your free copy of "10 Slick Tips for Improving Your Child's Study Habits," visit her website at http://myslicktips.com/

To Succeed - Simplify

Remember back to high school algebra, how we wondered what use the formulas and all that work would be in real life. Those quadratic equations were, in a word, intimidating. We challenged ourselves, maneuvered and manipulated, stewed and considered, reduced and pureed, until finally we could determine "x" or at least leave the equation of unknown variables in its simplest form. Have you used the equations much since then?

Well, the incite, the answer, the solution to all that time and struggle is not found in coefficients or factors. The life lesson of advanced math is in a concept - simplify. Math is a means for reducing complexity, in abstracts as in life. We succeed when we deal with life in manageable chunks and reduce multifaceted circumstances to their easiest form. To succeed - simplify.

Look at what is happening around us: the political turmoil, the financial madness, the burgeoning technology, the quiet and not so quiet pleas of desperation. Life seems to be accelerating, sweeping more and more people into its tumult. The population is growing rapidly. We are polluting the environment at an unprecedented pace. The people controlling the resources are fighting handily to ensure the apple cart is not upset and they seize ever more of the pie. We continue to shuffle the deck chairs as the ship sinks or strum the fiddle as the great city burns. While the financial titans, captains of industry and the pawns they control continue to belch out the smoke and readjust the mirrors the masses are left wondering: Who do we believe? Which way do we go?

The question has been asked, "Where have all the leaders gone?" To secure their fortunes and ensure their legacies, of course. This daunting task is not for "them". The task before us is ours to address - ours to solve. We are wasting our time, energy, talent, and ultimately our lives, fighting over the spilt milk when we must devote our assets: intellectual, financial and physical; to the substantial challenges that confront us. The issues, on scale, are all complex. Like quadratic equations they appear intimidating. However, we all lose when we allow complexity to overwhelm us. To succeed, we must simplify. And the simple answer, in the final analysis, comes down to relationship.

The biggest challenge of an individual life, reflected in the health of society at large, is to connect with other journeying souls. Unfortunately, for all the technology of modern life, from high definition television, to high tech automobiles and other forms of advanced transportation, to smart phones, and sophisticated computers, we keep making life more complex. Technology is being employed not as a means to a better life, but as a distraction - to entertain and manipulate. Even the young, who seem to be masters of the technology, are voting with their lives. Suicide is on the rise in the developed world. Complexity is quite literally killing us.

We are lost in the complexity of daily living when the task is simple - connect with other human beings. This connection is not a means to an end. The connection is an end in itself. Forget about the medium, forget about the media - simplify life - simplify your life. Connect with another human being - directly - in a place, in a moment. Your life will instantly become more balanced, more full and more fulfilling.

Don't allow the barrage of consumer messages, the pan handlers of material prosperity, to divert you from your course. You need this, society needs this, the world needs this. Simplify your life - spend quality time connecting with other people. Through your connections, through the bridges you build and the understanding you foster, you will change the world. And all of us, and the world, will be better for it. To succeed - simplify.

Copyright (c) 2010 Scott F Paradis

Scott F. Paradis, author of "Promise and Potential: A Life of Wisdom, Courage, Strength and Will" http://www.promiseandpotential.com publishes "Insights" and a free weekly ezine, "Money, Power and the True Path to Prosperity". Subscribe now at http://www.c-achieve.com

Algebra: How To Start and Why You Really Need to Practice a Lot

The most importand factor about algebra and mathematics is that you really need to practise in order to understand the concepts. There are a lot of algebra books, tutorials, interactive lessons, online tutors and courses that can help you a lot, but without some homework and practise you are bound to fail.

You cannot remember the thousands of algebra concepts and equations without truly understand what each equation really means. What is the reason behind every algebra concept.

A key factor is to take small steps. For example learn one algebra equation, practise it for a couple of days, then move on to the next. Move on only if you feel comfortable with what you've learned. Every time you move on to the next algebra concept increase the difficulty of the equations or the exercises you're trying to solve.

Follow this procedure and soon you'll notice that your brain will be able to absorb mathematics and algebra quicker and easier. You will start to like it.

Another thing is that algebra requires you to apply specific rules. You will find these basic rules in a lot of equations and algebra concepts. The more you practise the basic algebra rules the better you'll become at learning different and more difficult algebra concepts.

A lot of people learning algebra miss out on the basic rules. In the end they mess up everything. Remember, rules are everywhere. Our society is based on specific rules. Same goes for algebra. Treat algebra rules like a game and you'll like it.

You may end up wondering.. Why should I learn math and algebra? That's a good question. You must ask yourself this question before you go on. Algebra will help you solve problems, make decisions, apply strategies and a lot more. If you want your skills to worth gold then you have to learn math and especially algebra. If you want your mental powers to reach top level then you have to learn algebra.

If you like mathematics then you know that in order to move to advanced mathematics you have to know the basics of algebra. Algebra is everywhere in your everyday life. This is a great advantage when trying to learn algebra. Why? Because you can take the a real situation and turn it into an algebra concept or equation.

Always ask yourself.. why? Here is a simplified example. You walk on the street and you see a girl carrying a birdcage. Inside the birdcage there are 4 little birds. Suddenly the cage's door opens and one of the birds escapes. You can turn this incidence into a simple algebra equation:

3 - 1 = 2

See what I mean? Now that's the first step! After that it will be easy to turn live complicated situations into algebra concepts. Your brain will soon become familiar with the idea. And soon you will be able to turn your entire life into algebra!

If you are a starter then you have to take a look at this interactive math tutor that will help you master the concepts of mathematics quickly. You can try the free demo lessons. It will help you a lot to get started and boost your self esteem. These fine tuned interactive algebra e-courses will help you learn faster, easier, with a funny and entertaining approach.

Algebra for Dopes - It Ain't That Hard - The Quadratic Equation - Part II

As mentioned in Part I of this series, mastering algebra requires little more than clear-cut explanation and some concerted practice. Conquering the diverse aspects of this field necessitates focused thought and willingness to see through the trees to view the forest. Rather than view algebra-which encompasses many different topics and has vast extensions throughout-in one fell swoop, it is much better to view this subject piecemeal. This is the approach that will be taken in this series of articles.

Learning quadratic equations is one of the key plateaus in algebra as this opens the door to higher degree equations and even the calculus. One of the difficulties with this subject area is that the quadratic has many ways of being conquered, so to speak, that is to say, solved. Essentially there are three ways to solve a second-degree, or quadratic equation: one is by factoring; two is by completing the square; and three is by using the famous quadratic formula.

Now this is where it gets a little sticky since each of these three methods is a whole field within itself. Of the three methods, both the formula and the completing the square will always work on any quadratic equation. Factoring will only work on a subclass of the entire class of second-degree equations, and it is this class that we will discuss briefly here. Specifically, we will look at the subclass of factorable quadratics which have terms with all positive coefficients. That is, if we consider the general quadratic ax^2 + bx + c, then a, b, and c will all be positive. For the sake of classification, we will call this subclass of quadratics the "abc positives."

In order to solve an "abc positive" by factoring, it is necessary that the multiplicative factors of the c-term be able to combine additively or subtractively to the b-term. Now that this is starting to sound like a lot of mumbo jumbo, let's break this down and show how simple this really is. Let's do this with a specific example. Take x^2 + 7x + 12. The pair-wise factors of 12 are 1 and 12, 2 and 6, and 3 and 4. When we multiply any of these pair-wise factors we get 12. We look for the pair that combines additively or subtractively to give 7. It is obvious that only 3 and 4 do.

From this, we factor as such: x^2 + 7x + 12 = (x + 3)(x + 4). The solution to this quadratic is now the opposite of 3 and 4, or -3 and -4. That is, if we substitute -3 or -4 into the original quadratic equation, the result will be 0, for this is what it means for a number to solve a quadratic equation: it makes the equation 0 upon substitution. I leave it to the reader to verify that -3 and -4 do indeed do this.

After taking a look at one more, you will now have the secret to solving the entire subclass of abc positives. Take x^2 + 9x + 20. The pair-wise factors of 20 are 1-20, 2-10, and 4-5. The only pair that add to 9 are 4-5. Thus x^2 + 9x + 20 = (x + 4)(x + 5) and the solutions are -4 and -5. Once again, you can confirm that when each of these values is substituted back into the original quadratic, you will get 0.

When the vast subject of algebra is broken down to size, you start to see that it really is not as difficult as some might make it. In fact, you might start to see that it really is quite understandable. After looking at this approach, you come to realize that anyone could master this field. It really is true: algebra really ain't that bad. And once algebra is mastered, you can be thrilled to know that mathematics opens up a magical door which leads to a rich world of discovery.

See more at Cool Math Site and Cool Algebra Ebooks

Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Under the penname, JC Page, Joe authored Arithmetic Magic, the little classic on the ABC's of arithmetic. Joe is also author of the charming self-help ebook, Making a Good Impression Every Time: The Secret to Instant Popularity; the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)-particularly in regard to its educational flavor- continues to captivate readers and to earn him recognition.

Joe propagates his teaching philosophy through his articles and books and is dedicated.

Algebra - Masterminding Maths in the Easiest Way

When a child first steps into his kindergarten school room, he looks at the world with rose colored glasses. Hardly does he realize what is waiting for the next twenty years of his educational life. He is introduced to his alphabets first and starts to play with his 'A B C D'. In his days of playschool, it is beyond his imagination, how his A, B, C and X Y Z can cause him spending a chain of sleepless nights in a row before his mathematics examination. Mathematics is one subject which ends up terrorizing some of us (better to be read as many) throughout our student life. It is in fact amusing to note how some researches show the percentage of mathematics freaks being elevated among the fairer sex. With so much mathematical troubles especially in a difficult and intricate field like algebra, most of us need guidance in mathematics beyond the reach of classroom teaching.

All those people who are stricken by continued euphoria due to the problematic combination of numeric's and alphabets, should opt for the specialized coaching classes held for educational assistance in mathematics. Algebra mainly deals with the study of the relation between quantity and construction of variables by basic addition, subtraction, multiplication and division at its root frameworks. At a little more advanced stage, it also demands the factorization and roots of different numbers, polynomials and variables in question. The elementary classified segments in algebra are pre-algebra, linear-algebra, elementary algebra, abstract algebra and universal algebra.

The crucial fields of algebra which students require to specialize in are basic equations, polynomials, a single or a set of variables and fundamental arithmetic. The usually required for concepts are fundamental theorem, the quadratic, linear, cubic, polynomial, quintic and quadratic equation. Tutors for the most sought after topics of algebra are very easily found these days as competent professors and teachers and even people in other respectable positions are taking up tutoring as their part time earning. You can not only find tutors online, but also avail the algebra tuitions given out over the net. Even nowadays so many recorded tutorial CD's are also available in the market to help you learning algebra. So if you want to be an adept in Algebra it's time to improve your knowledge, this could be a private tutor or an online tutorial both can play a significant role to enhance your knowledge-base.

Search web sites for Experienced online algebra tutor. It is an easy and hassle free way to find good tutor online.

A B C of Solving Quadratic Equation

An expression of the type a x ² + b x + c = 0 , ( a ≠ 0 ) is called a quadratic equation in the variable x .

The equation a x ² + b x + c = 0 is called the general (or, standard form)

We can solve a quadratic equation by (1) factorization or by (2) applying the formula.

The formula of finding the roots of the quadratic equation is as follows

x = (- b ± √ (b ² - 4 a c) ) / 2 a

Now we will discuss how to solve applied problems. Due to wide variety of applied problems, there is no single solving technique that works in all cases. However the following suggestion proved helpful.

Step: 1 Read the problem carefully and determine what quantity (s) must be found.

Step: 2 assign a variable name to the quantity.

Step: 3 try expressing the problem algebraically, and as well determining which expressions are equal and write the necessary equation (s).

Step: 3 solve the resulting equation (s)

Now go through a simple problem based on formation of quadratic equation and solving

Problem: The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 58 / 21 . Find the fraction.

Solution:

Let the numerator of the fraction be x (x be an element of I )

Then its denominator is (2x + 1).

So the fraction is x / (2x + 1))

And the reciprocal would be (2 x + 1) / x

According to the problem:

(X / (2 xs + 1)) + ((2 x + 1) / x) = ( 58 / 21 )

(X ² + (2 x + 1) ²) / (x + (2x + 1)) = ( 58 / 21)

[L. C. D is = ( x + ( 2x + 1 ) ]

21 ( x ² + 4 x ² + 4 x + 1 ) = 58 x ( 2x + 1 )

105 x² + 84 x + 21 = 116 x ² + 58 x

11 x ² - 26 x - 21 = 0

11 x ² - (33 - 7) x - 21 = 0 [using middle term factorization]

11 x ²- 33 x + 7 x - 21 = 0

11 x ( x - 3 ) + 7 ( x - 3 ) = 0

(x - 3) (11 x + 7) = 0

Either ( x - 3 ) = 0 , or ( 11x + 7 ) = 0 [ using zero factor theorem ]

x = 3 ,

From, (11x + 7) = 0

We get, x = - (7 / 11)

But x is an integer , neglect x = - (7 / 11)

Take x = 3

So the required fraction, (x / (2x + 1)) = (3 / (2 * 3 + 1))

= (3 / 7)

Now try the following:

The age of a man is twice the square of the age of his son .Eight years hence the age of the man will be 4 years more than thrice the age of his son. Find their present age?

If you cannot solve this problem, you probably need more practice. A good online tutor would be helpful if you plan to master this subject in a short period. Any reasonably good math tutor should do.

Ishani Dutta is an educator who specializes in providing online tuition packages for Mathematics and English. For similar tips on different topics like pre-algebra, co-ordinate geometry, essay writing,english grammar, mailto:info@learningexpress.biz or go to: [http://www.learningexpress.biz] for online help

Quadratic Equations - An Introduction

Quadratic equations are the algebra topic taught to grade ten or eleven students. The word quadratic means, degree two in mathematics. Therefore any equation in degree two is called a quadratic equation. The form of standard quadratic equation is written as given below:

ax² + bx + c = 0

Where, "a", "b" and "c" are the real numbers and "a" can't be zero because in that case the quadratic term "ax²" becomes zero and the equation itself lose its identity and change to linear equation (degree one) which can be written as "bx + c = 0".

Some examples of quadratic equations are given below to make their identity more clear.

1. 3x² + 2x + 5 = 0

2. - x² + 3x - 9 = 0

3. x² + 1 = 0

4. - 9x² - 6x - 8 = 0

5. 4y² + 9 = 3y

Keep in mind that any letter can be used as a variable in the equations as I used "x" and "y" in the examples above.

In standard form these equations have three terms; first term in degree two called the quadratic term, second term in degree one called the linear term and third term is a constant number as shown in above examples.

Look at example # 3, there are only two terms in the equation. The term with degree one (linear term) is missing because the coefficient for this term is zero. This example can be written in standard form as shown below:

x² + 0x + 1 = 0

Now you have understood the way to write quadratic equations, the next step is to know about solving these equations. There are many ways to solve, such as solve by graphing, factor method, square root method, completing the square method and last but not least the formula method to solve quadratic equations.

To solve these equations using factoring method basic knowledge of factoring polynomials is required. You can read my articles about factoring polynomials for deeper knowledge about the topic.

To use formula to solve these equations, students should be very confident in radicals and they specially should have good knowledge of square roots. There is a special character used in formula called discriminate and is denoted by "D". The value of "D" is calculated by using the following formula:

D = b² - 4ac

Or in other words, linear coefficient "b" squared minus 4 times quadratic coefficient "a" times the constant term "c".

These equations if plotted on the graph, make a special cup shaped curve called parabola. There is a separate unit in grade eleven or twelve text books to study about parabolas.

There are many applications of these equations in higher algebra and to solve equations in higher degrees.

For more math resources and math worksheets my site can be visited or click for free 2nd grade math worksheets, for your kids in 2nd grade.

Why Algebra Looks So Hard?

What is algebra and what it does?

Algebra is a branch of mathematics. Algebra deals with the variable activities in our daily lives. It comes after learning the arithmetic in math.

What are the variable activities?

Variable means something that keeps on changing. Variable activities are the activities which don't stay the same over time. They keep on changing, for example; some trends can go up or down, left or right and east or west etc.

For example; the weight of a person never remains the same, it keeps on changing by getting lower or higher every day. Sun doesn't stay at the same spot whole day; it looks changing its position all the day along (it is because of motion of Earth around the Sun though). Share markets keep getting higher and lower on every single moment. A worker's pay changers according to number of hours he/she worked.

Finally, it can be said that algebra is study of activities which keep on changing with time. As we have hundreds of changing activities around us, therefore, algebra is everywhere in our daily lives.

What are the basic concepts need to be learned before starting algebra?

There is some basic Pre-Algebra concepts need to learned before starting algebra. These topics are given below;

Basic addition, subtraction, multiplication and division.
Times tables at least up to times by 10.
Know how to write all the factors of a number, finding greatest common factor (gcf) and least common multiple (LCM).
Fractions and operations on fractions.
Decimals
Integers
Order of operations

What are the main topics learned in algebra?

The main topics in algebra are;

Knowledge about the variables
Know the coefficients and constants
Writing algebraic expressions
Simple linear equations in one variable
Polynomials, degree and type
Operations with the polynomials, such as, adding polynomials, subtracting polynomials, multiplying and dividing polynomials
Rational expressions, Factorization
System of linear equations in two and three variables
Quadratic equations, absolute value equations and inequalities
Patterns in general like sequences and series.

Why is algebra hard?

Algebra is not that hard. One can take it as a challenging course. It looks hard as it is based on very very general terms.

Generic terms mean, for example; let's say you are looking for your younger brother, Vicky, in a crowd. You can call him by saying his name, Vicky aloud. Also your brother is a boy, and you can call him by saying, boy aloud. This time the problem is, there is too many boys in the crowd and may be, all of them start looking at you. So, boy is a general name and Vicky is a proper name.

Now if you recognize your brother from his clothes, height or other appearance you can even find him in the crowd by saying boy aloud. It looks hard, though but once your brother recognizes your voice, he will come to you. Same way you need to call algebra aloud (I mean learning it) and it will come to you like your brother.

Algebra has some rules to follow (as your brother has some unique appearance other than his name) and if you follow these rules, algebra is not hard at all.

So, that's all about basic algebra terminology, I could open in front of you according to my little knowledge of the topic.

For basic math content such as 2nd grade math, place value or basic fractions; our sites can be visited by clicking any of the links given in this article.

To learn fractions from basic fractions to adding fractions with mixed numbers or multiplying or dividing fractions lessons and worksheets, stay tuned as more helpful math tips and links to our math pages on its way.

Pre Algebra - A Prefix to the Study of Algebra

Algebra is an integral part of mathematics. It can be considered, a tough subject, if the basics are not clear. And to attend to this requirement of budding students, the education system of United States introduced the course of Pre Algebra. It is generally taught in schools in between seventh and ninth grades. However for practice it can be started from sixth grade too.

The main aim of Pre Algebra courses is to make the students develop a healthy sense of numerical relationships. The main emphasis of the teachers at this level falls on manipulating integers, fractions, percents, ratio and proportions decimals, working with variables; critical thinking, reading and comprehending graphs and general problem are solving tricks.

Algebra works as a major pillar towards understanding the crucial study of Algebra. It gives the foundation towards the main study. Algebra has now become important in schools and college studies. It hampers the ranking and the grade system to score bad in the subject. Similarly Algebra is important for getting good jobs and establishing a successful life. All the big jobs wants employees to be of good numeric and analytical skill and this can be achieved with the help of Algebra. It develops the sense of logic in human being which helps them to face the future complexities of life. However to attain this level the primary factor is the complete and thorough knowledge of Pre-Algebra. To solve the larger equations you need have a good idea on this.. It is important that you learn the tricks of Pre-Algebra from an experienced teacher who can guide you well on the subject.

In every field of life, Pre-Algebra and Algebra are important. If you become a scientist, there is absolute no way to escape the study of Pre-Algebra. Even if you are working in a small firm or a big construction company you need to master the field of algebra.

A good foundation is important to learn to manipulate equations later.

Timcy Hood writes informative and unique articles about Homework help and Algebra 1. One thing that separates Timcy Hood from others is the passion she puts into it, knowing full well and respecting the time viewers spend on reading his work.

Algebra Made Easier With Algebra 2 Homework Help!

Algebra 2 is a step beyond Algebra 1. Before commencing this stage in the educational process, students need to be thoroughly grounded with the foundations. Algebra 2 homework help can answer some of the basic questions about Algebra. But is it advisable that students have tried to do their personal Algebra homework before consulting with this source. In Algebra 2, students are introduced to logarithms and exponents, graphic functions, ways of solving inequalities and equations with complex numbers. The course structure also includes polynomial arithmetic, rational expressions, radicals and complex numbers, quadratic system and cone sections. As new terms, these words might sound a little complicated. However, Algebra 2 homework help becomes a personal guide to make complex mathematics easier for students.

Help for competitive examinations

Algebra 2 homework help is a very effective platform when preparing for competitive examinations or a university entrance examination. A foundation in this category of mathematics is a must when a student is applying for the General Educational Development examination. The GED exam earns the taker the equivalent of a high school diploma, which is necessary for students who cannot complete their high school courses. The university entrance exams are the SAT and ACT. Students need not bother to seek out a personal teacher. All the queries are easily resolved with the resources found at the Algebra 2 homework help site.

Benefits of the Algebra 2 Homework help platform

This platform's motto is to make mathematics easier for students. By availing themselves of this service, students can get rid of the complexities of the subject. Algebra 2 homework help provides tips for the effective study of mathematics. The students have to possess sharp memory skills in order to excel in this section of math. The help book provides certain very interesting ways of making memory skills sharper to assist the learning process. This is also helpful for students who do not remember the elementary lessons of Algebra. It is not possible to provide a detailed version here, but a basic reference is provided. This reference helps students brush up their memories to remind them of the lessons that are learned in Algebra 1.

Algebra 2 homework help trains students to first understand the problem and then locate the most efficient way to solve it. For instance, when looking to solve an equation, the first step is to observe the number of terms in the given equation. The next step, then, is to decide which type of factoring to opt for to solve the equation. Algebra 2 homework help also works on a student's critical thinking capability. This process helps in case students want to take up mathematics for further higher studies. It helps students gather a sound knowledge required in order to deal with complex mathematical problems at higher levels. It is as easy and friendly as troubleshooting.

Career prospects made attractive with Algebra 2

People who excel in Algebra have great career prospects. They might not be aware of this, but they can get really high salaried jobs. They can even join any educational institute and spread their knowledge.

Are you facing problem in your algebra 2 course work? Are you looking for homework help live, so that you can understand your topics better in the comforts of your home? Visit Thank you Tutor, to get the best Algebra 2 homework help!

Why Study Algebra? Completing the Square It's Not As Hard As You Think

One of the most useful techniques in algebra is that of completing the square. The name is appropriate as the geometric interpretation encompasses the formation of a square from a rectangle by the addition of an appropriate quantity. Geometry aside, this technique has many applications, not only in algebra, but also in more advanced realms such as integration, which is a key component of integral calculus. Here we will see that this technique can be had rather inexpensively.

Completing the square involves taking a non-perfect square trinomial and converting it into a perfect square. Actually, this technique is performed when you have a quadratic equation set to zero, as in x^2 + 10x - 5 = 0. If you recall, a perfect square trinomial is one in which the middle coefficient is equal to twice the product of the square roots of both the leading coefficient and the constant term. What a mouthful! Let's look at a specific example. Take the quadratic trinomial x^2 + 10x + 25. The leading coefficient is 1, the number (which is understood) in front of the x^2 term. The middle coefficient is 10, and the constant term is 25. The square root of 1 is naturally 1; the square root of 25 is 5; 2*1*5 is 10, which is the middle coefficient. Thus x^2 + 10x + 25 qualifies as a perfect square trinomial.

So what is so special about these trinomials? Well for one, they can always be factored into the form (x +/- c)^2. In other words, we can always factor them as (x + c)^2 or (x - c)^2, where c is a constant and the "+" or "-" is dictated by the sign of the middle coefficient. Once factored, we can easily solve any quadratic equation by performing the simple operation of taking the square root and adding or subtracting the constant c. To see this, let us look at a specific example.

Suppose we wish to solve the quadratic equation x^2 + 8x - 10 = 0. You cannot solve this by factoring. You can of course go directly to the quadratic formula, but an even quicker way is to complete the square, and this is how we shall do it. Isolate the x-terms, namely x^2 and 8x, on one side of the equation and bring the constant term to the other. Remember that when we move the -10 over we get +10. Thus we have x^2 + 8x = 10. Now begin the process of converting x^2 + 8x into a perfect square. We take half of 8, which is 4 and square it to get 16. We add this quantity to both sides of the equation to get x^2 + 8x + 16 = 10 + 16 = 26. Now if you check the conditions which make a trinomial perfect, you will see that x^2 + 8x + 16 fits the bill. That is 2*4*1 = 8.

Since the trinomial is now perfect, we can factor it into (x + 4)^2, that is we take the x term, half of 8, and the "+" sign, since the middle term is positive. We write (x + 4)^2 = 26. To solve this equation, we simply take the square root of both sides, remembering to take the "+" and "-" part. (Remember: when we take a square root in an equation, we always consider both the positive and negative values). Thus we have (x + 4) = +/- the square root of 26. (Since I cannot use the square root symbol in this article, I will write 26^.5 as the square root of 26; actually this is true since the square root is the one-half power.) To finish this off, we subtract the 4 from both sides to solve for x, and we get x = -4 +/- (26)^.5, that isx = -4 + (26)^.5 or x = -4 - (26)^.5. Since (26)^.5 is equal to a little more than 5, about 5.1, we have that x is equal to about 1.1 or -9.1.

With this technique, you can now solve any quadratic, regardless of whether it is factorable or not, without resorting to the quadratic formula. To sum up, all you need do is the following (As you read these steps, refer back to the example just done):

1) Isolate the x terms on one side of the equation and the constant term on the other;

2) Take half the middle coefficient, square it and add it to both sides of the equation;

3) Factor the trinomial using (x +/- c)^2, where c is equal to half the middle term, and the sign is taken according to the sign of the middle coefficient; and

4) Take the square root of both sides, remembering to consider the +/- cases, and add or subtract c to both sides.

With the ammunition given above, you are now expert at completing the square and solving any quadratic equation. Isn't life grand!

Joe is a prolific writer of self-help and educational material and is the creator and author of over a dozen books and ebooks which have been read throughout the world. He is a former teacher of high school and college mathematics and has recently returned as a professor of mathematics at a local community college in New Jersey.

Joe propagates his Wiz Kid Teaching Philosophy through his writings and lectures and loves to turn "math-haters" into "math-lovers." See his website http://www.mathbyjoe.com for more information and for testimonials, and try out one of his ebooks here http://www.mathbyjoe.com/page/page/2924777.htm to achieve better grades in math.

Algebra for Dopes - It Ain't That Hard - The Quadratic Equation - Part III

In Part III of this series, we examine how we solve factorable quadratics of the form ax^2 - bx + c, in which both the a and c terms are positive, but the b, or middle term, is negative. Such an example would be x^2 - 5x + 6. This class of quadratics, which we will call the "b-negatives" are easily solvable using the factoring techniques we learned about in Part II.

To get right to the chase, let's look at the quadratic in the first paragraph: x^2 -5x + 6. Remember, we are looking for the solutions, or those values of x which when plugged in for this variable make the quadratic zero. This is another way of saying that those values are the zeros of the function, or those values are the ones which make the quadratic vanish. As in Part II, we look to see the factors of the c-term, which in this example is 6. They are: 1-6, and 2-3. Now this is a very good example because of the nature of these two factor pairs. You see, both 1-6 and 2-3 can combine in some way to give 5; for 1 and 6 can be combined subtractively, that is 6-1 = 5; and 2 and 3 can be combined additively, that is 2 + 3 = 5.

To decide which pair works, remember that this article treats the b-negative class of quadratics. Since the c-term, or 6 is obtained from multiplying the factor pair and only two positives or two negatives can produce a positive result, and since the b-term or -5 can only be obtained by adding two negative numbers, the only factor pair that works is 2-3. For if it were 1-6, then -1 and -6 add to -7 not -5. Thus x^2 - 5x + 6 = (x - 3)(x - 2), and the solutions are 3 and 2. The reader can easily verify that either of these values substituted for x in the original quadratic makes the equation vanish.

To see how this method works with one more example, let's try x^2 - 10x + 16. The factor pairs of 16 are 1-16, 2-8, and 4-4. The only pair that will combine to give 10 is 2-8. Since we have a b-negative quadratic we must have both of these numbers as negatives. Therefore, the quadratic factors into (x - 8)(x - 2) and the solutions are 8 and 2. Nothing at all difficult about this.

If you have followed this series, you are starting to see how we take algebra down to size. By "chunking" this subject up into little component pieces, we make it a much more manageable beast. Thus far we have dissected the class of factorable quadratics into two subclasses, each of which is very easy to deal with. Once this view is taken, the rest of algebra becomes just as manageable and much more pleasant.

See more at Cool Math Site and Cool Algebra Ebooks

Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Under the penname, JC Page, Joe authored Arithmetic Magic, the little classic on the ABC's of arithmetic. Joe is also author of the charming self-help ebook, Making a Good Impression Every Time: The Secret to Instant Popularity; the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)-particularly in regard to its educational flavor- continues to captivate readers and to earn him recognition.

Joe propagates his teaching philosophy through his articles and books and is dedicated.

Math Homework Help - How to Easily Identify and Solve Quadratic Equations

One of the most common questions that a student asks his or her algebra tutor when seeking math homework help concerns finding math solutions for problems involving quadratic equations. Before attempting to solve any equation, the algebra tutor should aid the student in identifying this type of equation. It can easily be identified by the highest power of the variable x, which should be equal to two. When math solutions require the student to solve a quadratic equation, the algebra tutor should focus on how to solve the equation for the value(s) of x when y is set equal to zero. In other words, the student should solve for the x-intercept(s). The x-intercept(s) are the point(s) at which the graph of the quadratic equation cross(es) the x-axis. Alternatively, the student may be asked to find the zeros or the roots of the quadratic equation, which are identical to solving for the x-intercepts! There are several different ways in which the student can solve this type of equation. Firstly though, y should be set equal to zero. Once this is accomplished, the equation can be solved using either graphing, factoring, or using the quadratic equation.

When providing math homework help, the algebra tutor should highlight that the least accurate method of solving the equation involves graphing the equation and noting where the graph crosses the x-axis. These points are referred to as the x-intercepts as mentioned before. Note that there may be either zero, one, or two x-intercepts. The math solutions for this type of problem are usually not listed as points, but rather as values of x. This method may potentially yield inaccurate solutions since it involves reading values off of a graph that may not have been drawn with complete precision by the student. In order to correct this problem, the student may also use a graphing calculator to check his or her math solutions.

Factoring is another, more exact method that can be used by a student seeking math homework help to solve a quadratic equation. From the start, the algebra tutor should emphasize that not all quadratic equations are factorable. For that reason, it is always a good idea for the student to as well be familiar with using the quadratic formula which will be discussed shortly. Factoring can be useful since it is quick and can easily be checked by plugging the solutions back into the original quadratic equation.

The last method to be discussed is the quadratic formula. This method is foolproof in that the student does not necessarily need to know how to factor the original quadratic equation. Also, this method allows the student to solve for x-intercepts that are not necessary whole numbers. In other words, in terms of math homework help geared toward the student, the quadratic equation can be used to solve for radical, irrational, or even imaginary solutions! The algebra tutor should as well help the student realize that the quadratic formula can only be used to find solutions when the original equation is in general (or standard) form. This means that the quadratic equation cannot be in vertex form. If this is the case, the quadratic equation can easily be converted to general form so the quadratic formula can be used. In the quadratic formula, a represents the coefficient of the term with the x-squared term, b represents the linear coefficient, and c represents the constant term (the term with no variable multiplied onto it). Once these are identified, the quadratic formula can easily be used to find math solutions for a variety of different problems involving equations.

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Do You Need Help With Algebra Homework?

If you are a student of Algebra then many times you must have been in situations when you need help with your homework, but there is no one you can turn to. When learning algebra, it is important to have a clear understanding about the various math definitions and rules such as exponents, graphs, factoring quadratic equations, and the quadratic formula. Many times class instruction may not be enough as there are so many students in a class that teacher cannot pay attention to all. The slow learners usually suffer in a classroom setting. If you are one of those then don't live with the fear of math as there are many expert algebra tutors available to help you with homework.

There are several free algebra websites that provide help with homework. This can be a good method to deal with the problem, but this is not a long term solution. Most of these websites offer free homework solutions with the help of formulas, worksheets, practice tests, and quizzes. You can also post your query or problem on a forum, blog or message board. However, there is a disadvantage in doing so as you may have to wait for several days to get an answer. Another drawback is that the answer may not be sufficient enough to help you understand the concept clearly. To understand a problem in the right way, you must understand the method of solving the problem.

Though free algebra homework answers are available online, it is important for a student to get a clear understanding of the different topics of math. Another alternative method would be to seek help from friends and family members. If there is someone in the family who is good in algebra then he or she can help you with your homework. Sometime, you may need help with just a specific topic of algebra, and most students seem to seek help in factoring.

You can seek help from the above mentioned methods to solve an algebra problem but this may not be enough. Thus, consider hiring a private algebra tutor online. Thankfully, there are several websites that provide a list of private tutors available in your area. You can now easily save time and trouble by seeking help from a professional algebra tutor in your neighborhood. Private tutoring services have several benefits. You will not have to travel to any far off place to get your doubts cleared. An expert algebra tutor can help you with homework and clear all your doubts related to the subject.

Do you need help with Algebra homework? You can now hire a private algebra tutor in your locality today. Search for private algebra tutor in San Diego, now.

Algebra DVDs - An Excellent Learning Tool to Teach Your Kids Algebra

Algebra is the line of demarcation between basic and higher math. Many children panic upon seeing letters and numbers in a mathematical equation. They often run to their parents for help. Parents there is a resource specifically designed to help you teach your kids algebra, Algebra DVDs.

Algebra DVDs can be a parent's best friend. They provide simple step by step instructions on the basics of algebra along with simple exercises allowing parent and child to test their mastery of each topic. The taped lessons use cartoons, songs, and real world examples to reinforce the concepts being taught.

One of the greatest gifts a parent can give a child is to show them your willingness to learn even though you are an adult. This helps children to understand that not knowing something is not bad as long as you are willing to admit you don't know and seek out help.

A parent and child working their way through this teaching tool together can help to strengthen their bond while increasing their mastery of the subject matter.

Mastering mathematics requires a solid understanding of Algebra. Scientist, engineers, business people and industrialists use algebra to solve problems every day. Understanding Algebra gives children more career choices.

Parents can use the guided lessons to help their children understand basic terms like variable, equation, and factors and follow them through the more complex concepts.

One of the great strengths of the Algebra DVDs is the ability to replay them endlessly until you understand a concept. Standard classroom instruction does not allow you to do this. So buy Algebra DVDs, teach your child algebra and watch their confidence grow.

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Algebra For Dopes - It Ain't That Hard - The Quadratic Equation - Part V

In Part V of this series, we examine how we solve the last class of factorable quadratics of the form ax^2 + bx - c, in which the b-term is positive and the c-term is negative. Such an example would be x^2 + 4x - 5. This subclass of quadratics are as easily solvable as those of the "bc-negative" class discussed in Part IV of this series.

To show how similar this class is, let's examine x^2 + 4x -5. This is the same quadratic as the first example in Part IV of this series, except the 4x term now is positive instead of negative. As in the last article, we note that 5 is prime and its only factors are 1 and 5. Since the c-term is negative, the 1 and 5 must be of opposite signs. Since the b-term is positive, the larger number must bear the positive sign; otherwise the result of the b-term would be negative. Thus x^2 + 4x - 5 = (x + 5)(x - 1), and the solutions are -5 and 1.

Although this method should be perfectly clear by now, let's reinforce it with two more examples. Let's take the quadratic x^2 + 10x - 24. The factor pairs of 24 are 1-24, 2-12, 3-8, and 4-6. Notice that as the c-term becomes a larger composite number as in this case, generally the number of possible factor pairs increases. When the c-term is a prime number, as in the first example, then the only factor pairs are 1 and the number itself. (By the way the first 10 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29; 2 is the one and only even prime; 1 is not considered a prime number.) Now of the four factor pairs, the only one that can combine to give 10 is 2-12. Since 10 must be positive and 2 and 12 must be of opposite sign, can you guess which must be positive and which negative? Easy enough, right? Thus x^2 + 10x - 24 = (x + 12)(x - 2) and the solutions are -12 and 2.

Finally, we will solve x^2 + 31x - 66. The factor pairs of 66 are 1-66, 2-33, 3-22, and 6-11. The only pair that combines to yield 31 is 2-33. Again, using the argument above this quadratic must factor as (x + 33)(x - 2), and the solutions are -33 and 2. The reader can easily verify that both -33 and 2 are in fact the zeros of this particular quadratic.

After following this series of articles, you are starting to see how quick you can become at algebra once you understand the rules of the game. And this goes for all of algebra: as we break down each component of this subject and apply these techniques, algebra-and indeed math-no longer is a mystery that perplexes, but a mystery that both enriches and enlightens.

See more at Cool Math Site and Cool Algebra Ebooks

Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Under the penname, JC Page, Joe authored Arithmetic Magic, the little classic on the ABC's of arithmetic. Joe is also author of the charming self-help ebook, Making a Good Impression Every Time: The Secret to Instant Popularity; the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)-particularly in regard to its educational flavor- continues to captivate readers and to earn him recognition.

Joe propagates his teaching philosophy through his articles and books and is dedicated to helping educate children living in impoverished countries. Toward this end, he donates a portion of the proceeds from the sale of every ebook. For more information go to http://www.mathbyjoe.com

Factoring Trinomials (Quadratics) - Lucid Explanation of the Method With Examples

Consider the product of the two linear expressions (y+a) and (y+b).

(y+a)(y+b) = y(y+b) + a(y+b) = y^2 + by + ay + ab = y^2 + y(a+b) + ab

We can write it as

y^2 + y(a+b) + ab = (y+a)(y+b) .......(i)

Similarly, Consider the product of the two linear expressions (ay+b) and (cy+d).

(ay+b)(cy+d) = ay(cy+d) + b(cy+d) = acy^2 + ady + bcy + bd = acy^2 + y(ad+bc) + bd

We can write it as

acy^2 + y(ad+bc) + bd = (ay+b)(cy+d) .......(ii)

Equation (i) is Simple Quadratic Polynomial expressed as Product of Two linear Factors and Equation (ii) is General Quadratic Polynomial expressed as Product of Two linear Factors

Observing the two Formulas, leads us to the method of Factorization of Quadratic Expressions.

In Equation (i),

the product of coefficient of y^2 and the constant term = ab

and the coefficient of y = a+b = sum of the factors of ab

Similarly, In Equation (ii),

the product of coefficient of y^2 and the constant term = (ac)(bd) = (ad)(bc)

and the coefficient of y = (ad+bc) = sum of the factors of acbd

So, if we can resolve the product of y^2 and the constant term into product of two factors in such a way that their sum is equal to the coefficient of y, then we can factorize the quadratic expression.

We discuss the steps involved in the method and apply it to solve a number of problems.

Method of Factoring Trinomials (Quadratics) :



Step 1 :

Multiply the coefficient of y^2 by the constant term.

Step 2 :

Resolve this product into two factors such that their sum is the coefficient of y

Step 3 :

Rewrite the y term as the sum of two terms with these factors as coefficients.

Step 4 :

Then take the common factor in the first two terms and the last two terms.

Step 5 :

Then take the common factor from the two terms thus formed.

What you get in step 5 is the product of the required two factors.

The method will be clear by the following Solved Examples.

The examples are so chosen that all the models are covered.

Example 1 :

Factorize 9y^2 + 26y + 16

Solution :

Let P = 9y^2 + 26y + 16

Now, follow the five steps listed above.

Step 1:

(Coefficient of y^2) x (constant term) = 9 x 16 = 144

Step 2:

We have to express 144 as two factors whose sum = coefficient of x = 26;

144 = 2 x 72 = 2 x 2 x 36 = 2 x 2 x 2 x 18 = 8 x 18; (8 + 18 = 26)

Step 3:

P = 9y^2 + 26y + 16 = 9y^2 + 8y + 18y + 16

Step 4:

P = y(9y + 8) + 2(9y + 8)

Step 5:

P = (9y + 8)(y + 2)

Thus, 9y^2 + 26y + 16 = (9y + 8)(y + 2) Ans.

Example 2 :

Factorize y^2 + 7y - 78

Solution :

Let P = y^2 + 7y - 78

Now, follow the five steps listed above.

Step 1:

(Coefficient of y^2) x (constant term) = 1 x -78 = -78

Step 2:

We have to express -78 as two factors whose sum = coefficient of y = 7 ;

-78 = -2 x 39 = -2 x 3 x 13 = -6 x 13; (-6 + 13 = 7)

Step 3:

P = y^2 + 7y - 78 = y^2 - 6y + 13y - 78

Step 4:

P = y(y - 6) + 13(y - 6)

Step 5:

P = (y - 6)(y + 13)

Thus, y^2 + 7y - 78 = (y - 6)(y + 13) Ans.

Example 3 :

Factorize 4y^2 - 5y + 1

Solution :

Let P = 4y^2 - 5y + 1

Now, follow the five steps listed above.

Step 1:

(Coefficient of y^2) x (constant term) = 4 x 1 = 4

Step 2:

We have to express 4 as two factors whose sum = coefficient of y = -5 ;

4 = 4 x 1 = -4 x -1; [(-4) + (-1) = -5]

Step 3:

P = 4y^2 - 5y + 1 = 4y^2 - 4y - y + 1

Step 4:

P = 4y(y - 1) - 1(y - 1)

Step 5:

P = (y - 1)(4y - 1)

Thus, 4y^2 - 5y + 1 = (y - 1)(4y - 1) Ans.

Example 4 :

Factorize 3y^2 - 17y - 20

Solution :

Let P = 3y^2 - 17y - 20

Now, follow the five steps listed above.

Step 1:

Coefficient of y^2 x constant term = 3 x -20 = -60

Step 2:

We have to express -60 as two factors whose sum = coefficient of x = -17 ;

-60 = -20 x 3; (-20 + 3 = -17)

Step 3:

P = 3y^2 - 17y - 20 = 3y^2 - 20y + 3y - 20

Step 4:

P = y(3y - 20) + 1(3y - 20)

Step 5:

P = (3y - 20)(y + 1)

Thus, 3y^2 - 17y - 20 = (3y - 20)(y + 1) Ans.

Example 5 :



Factorize 2 - 5y - 18y^2

Solution :

Let P = 2 - 5y - 18y^2 = -18y^2 - 5y + 2

Now, follow the five steps listed above.

Step 1:

(Coefficient of y^2) x (constant term) = -18 x 2 = -36

Step 2:

We have to express -36 as two factors whose sum = coefficient of y = -5 ;

-36 = -2 x 18 = -2 x 2 x 9 = 4 x -9; [4 + (-9) = -5]

Step 3:

P = -18y^2 - 5y + 2 = -18y^2 + 4y - 9y + 2

Step 4:

P = 2y(-9y + 2) + 1(-9y + 2)

Step 5:

P = (-9y + 2)(2y + 1)

Thus, 2 - 5y - 18y^2 = (-9y + 2)(2y + 1) Ans.

Example 6 :

Factorize (y^2 + y)^2 -18(y^2 + y) + 72

Solution :

Let P = (y^2 + y)^2 -18(y^2 + y) + 72

Put (y^2 + y) = t; Then P = t^2 -18t + 72

Now, follow the five steps listed above.

Step 1:

(Coefficient of t^2) x (constant term) = 1 x 72 = 72

Step 2:

We have to express 72 as two factors whose sum = coefficient of t = -18 ;

72 = 12 x 6 = -12 x -6; [(-12) + (-6) = -18]

Step 3:

P = t^2 -18t + 72 = t^2 - 12t - 6t + 72

Step 4:

P = t(t - 12) - 6(t - 12)

Step 5:

P = (t - 12)(t - 6)

But t = (y^2 + y);

So, P = (t - 12)(t - 6) = (y^2 + y - 12)(y^2 + y - 6)

In each of these two brackets, there is a Quadratic Polynomial which can be factorised using the five steps above.

y^2 + y - 12 = y^2 + 4y - 3y - 12 = y(y + 4) - 3(y + 4) = (y + 4)(y - 3)

y^2 + y - 6 = y^2 + 3y - 2y - 6 = y(y + 3) - 2(y + 3) = (y + 3)(y - 2)

See how these two Quadratic Polynomials are factorised with the knowledge of the 5 steps.

You might have mastered the 5 steps of factorisation by this time, to write directly like this.

Thus,

P = (y^2 + y)^2 -18(y^2 + y) + 72

= (y^2 + y - 12)(y^2 + y - 6)

= (y + 4)(y - 3)(y + 3)(y - 2) Ans.

For more, on Factoring Quadratics, go to http://www.math-help-ace.com/Factoring-Trinomials.html

Name : KVLN Age : 47 years old Qualifications : B.Tech., M.S. (from IIT, Madras) Has 14 years of teaching experience. Loves math and chess. Winner of state rank in the mathematical olympiad. University level chess player. Love for math and love for teaching makes him feel more than happy to help. For First-Rate Math Help, go to the author's web site http://www.math-help-ace.com/. It Aims to help to make every one an ace (expert) in math. Explains lucidly math topics for kids and teens with solved examples and exercises. Highlights the salient points and formulas. Helps to develop confidence and desire to continue. Helps to perceive the work as less demanding. Helps to complete their math home work more quickly. Helps to perceive the work as less demanding. Helps to to integrate the current problem with existing knowledge and ideas. Helps to encourage them to reach a solution on their own, with their active mental participation. Helps every student succeed in math by making the journey a pleasant one. The topics you can learn and enjoy include basic algebra and other algebra topics such as Equations, Inequalities, Polynomials, Factoring, Exponents, Logarithms etc. The pleasant journey includes Number Systems and other Numbers topics such as Divisibilty Rules, Prime Factorization, G.C.F., L.C.M., Prime Numbers, Perfect Numbers, Whole Numbers, Integers, Fractions, Decimals, Rational Numbers, Irrational Numbers, Real Numbers etc. Basic operations on numbers such as addition and subtraction including a number of solved examples and Exercises, multiplication including Multiplication Tables and Division including Long Division are lucidly explained. Math Word problems on Addition, Subtraction, Multiplication, Division, G.C.F., L.C.M., Linear Equations in one and Two Variables, Quadratic Equations etc., are lucidly explained.