Formulas are an important part of all math classes because they state relationships that are ALWAYS true, and they generally make various mathematical tasks easier to perform. Factoring is one of those fundamental tasks in Algebra. Factoring allows us to reduce algebraic fractions into simpler form, and it can help us solve equations. Factoring the difference of two squares is one of the most commonly used processes in all of Algebra. Understanding when and how to use it is critical to success in Algebra.
We have already learned the meaning of "to factor," but it is always a good idea to review the definition. Factoring is the process of re-writing an expression using multiplication.
Before we can factor the difference of two squares, we need to be able to identify it. What exactly is a difference of two squares? To fully understand, let's look at each word. "Difference" means subtraction, but subtraction of what? "Two" tells us that we have two numbers and/or algebraic expressions. Thus far, we know we are going to subtract one number or expression from another; but these numbers are special. Our two numbers or expressions are perfect squares, like 1, 4, 9, 16, 25, 36, 49, etc and/or a^2, b^4, x^2, (xy)^2, etc. A "difference of two squares" will look like 25 - 9 or x^2 - y^4. Now, we are ready for the actual formula.
In symbols: a^2 - b^2 = (a + b)(a - b)
In words: The difference of the squares of two numbers factors as the product of the sum and difference of those numbers.
Note: It is extremely important that you be able to state these definitions out loud and that you understand every word. Don't move on until you know you are ready.
Before we actually use this formula, let's make certain it is true. While this is not a formal proof, we are going to test this formula with a number example like 25 - 9. (Both 25 and 9 are perfect squares.) By our formula, since 25 = 5^2 and 9 = 3^2, 25 - 9 must be equal to (5 + 3)(5 - 3). So, is the formula true? 25 - 9 = 16 by just doing order of operations. (5 + 3)(5 - 3) = (8)(2) = 16. Both expressions have the value 16. Again, I caution that this is not a proof. Since the proof is not the point of this article, I will ask that you either trust me or do several more examples to convince yourself of the validity of this formula.
About now, you ought to be thinking, "Why would I want to do that?" It is easier to evaluate 25 - 9 than it is to evaluate (5 + 3)(5 - 3); but keep in mind that we will primarily be using this relationship for the purpose of reducing algebraic fractions and solving algebraic equations.
For example: Solve the equation x^2 = 16.
Many students will quickly jump to the "answer" of 4 since 4^2 is 16. However, this equation has two answers, but it is not obvious where the other answer comes from. Noticing that both x^2 and 16 are perfect squares, we should think about the possibility of a difference of two squares. We can rewrite x^2 = 16 as x^2 - 16 = 0. Now we have a difference of two squares that factors as (x + 4)(x - 4) = 0. The two different factors produce the two solutions, x = 4 and x = -4.
To make best use of this strategy, you must form a new habit. Every time you encounter an equation of the form x^2 = a number, take the time to re-write that equation. Thus, you must see a^2 = 121 as a^2 - 121 = 0. This, then, can be factored as (a + 11)(a - 11) = 0 for the two solutions of a = +11 and a = -11.
Forming a habit of constantly looking for a difference of two squares can make reducing algebraic fractions a much simpler process.
For example: If possible, reduce the fraction (x^2 - 25) / (x + 5).
We can recognize a difference of two squares in the numerator, so factoring it should be automatic. This produces (x + 5)(x - 5) / ( x + 5). Reducing the common factors of (x + 5) leaves the final reduced result of x - 5.
Why does it matter that x - 5 is the reduced version of (x^2 - 25) / (x + 5)? There are two different reasons for doing this. First, let's look at these two equations: (x^2 - 25) / (x + 5) = 12 and x - 5 = 12. These two equations are equivalent, but which one would you rather solve? Could you do the first one in your head? Probably not. But the second equation has the obvious answer of x = 17.
The second reason for using the difference of two squares is that it makes evaluation of expressions for a given value much simpler. We already know that (x^2 - 25) / (x + 5 ) is the same as x - 5. Now, let's pretend that x has the value of 13 and I ask you to evaluate (x^2 - 25) / (x + 5) for x = 13. The unsimplified version becomes (13^2 - 25) / (13 + 5) or (169 - 25) / (13 + 5) or 144/18 = 8. That was a lot of work. But evaluating x - 5 with x = 13 is simple: 13 - 5 = 8.
With only a little practice, you will be able to simplify and evaluate expressions like the one above in your head. The ability to recognize and factor the difference of two squares will make your life in Algebra class much easier.
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