What are Perfect Squares?
Perfect squares are the result of multiplying a number or variable by itself.
For example, 4 and 9 are perfect squares, because they are the products of two times two and three times three, respectively.
2 × 2 = 4
3 × 3 = 9
To see the free examples, please go to the next section of this page.
Perfect Squares in Algebra
In algebra, perfect squares are the product of two equal variables.
For instance, a2 and b2 are examples of this, because they are the products of a times a and b times b, respectively.
a × a = a2
b × b = b2
Binomials are two term expressions.
So, the expression a2 + b2 is an example of a binomial.
One type of binomial is the difference between two perfect squares.
The expression a2 – b2 is an example of the difference between two perfect squares.
Factoring the Difference of Two Perfect Squares
If the two terms of a binomial are perfect squares and the terms are subtracted, the binomial can be factored.
The factored form is as follows:
a2 – b2 = (a + b)(a – b)
In other words, the variables in the first set of parentheses will be added to each other and the variables in the second set of parentheses will be subtracted from each other when factoring the difference between two perfect squares.
We can prove the form above by multiplying each of the terms and canceling out as shown below.
a2 – b2 =
(a + b)(a – b) =
(a × a) + (a × –b) + (b × a) + (b × –b) =
a2 + –ab + ab + –b2 =
a2 – ab + ab – b2 =
a2 – ab + ab – b2 =
a2 – b2
In order to perform the factoring, you will need to be acquainted with the laws of square roots.
You will also need to understand how to do the FOIL method on quadratics.
You may also want to try our other problems on factoring and algebra.