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.

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.

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.

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!