Difference of Squares Factoring Guide

Factoring is a method that allows to see expressions in a different way or form. A factored form can allow us to simplify expressions which can help us in finding solutions to equations.

If we are given the form a^{2} - b^2, we can express this as a factored form as follows:

\displaystyle (a^{2} - b^2) = (a + b)(a - b)

If we were to verify this by going in the reverse way by expanding (a + b)(a - b), the calculations would be like this:

\displaystyle (a + b)(a - b) = a^2 - ab + ab - b^2 = a^{2} - b^2


Examples

Let’s demonstrate this factoring method through many examples.

Example 1

Factor x^2 - 9.

Solution:

We have a^2 = x^2 and b^2 = 9. So a = x and the positive part of the square root of 9 is 3 so b = 3. The factored form of x^2 - 9 is simply (x + 3)(x - 3).


Example 2

Solve the equation x^2 - 144 = 0.

Solution:

In this one, it wants you to solve the equation. In other words, find x-values such that x^2 - 144 = 0. To do this, we factor first and solve for the x-values from the factors.

We have a^2 = x^2 and b^2 = 144. So a = x and the positive part of the square root of 144 is 12 so b = 12. The factored form of x^2 - 144 is simply (x + 12)(x - 12).

From x^2 - 144 = 0, we have (x + 12)(x - 12) = 0. Now we equate each factor to zero and solve for x.
In (x + 12) = 0, x = -12 and in (x - 12) = 0, x = 12. Our solutions for x are 12 and -12.


Example 3

Factor x^4 - 16.

Solution:

Source: http://quicklatex.com/cache3/6f/ql_7d234bca80bcf63eaeb505bcce15ea6f_l3.png

This one does not look as obvious but if you know your exponent laws well, x^4 - 16 can be expressed as (x^2)^2 - (4)^2.

Here we have a^2 = x^4 and b^2 = 16. The factored form of x^4 - 16 is (x^2 + 4)(x^2 - 4).

Notice that we can factor further by factoring (x^2 - 4) to (x + 2)(x - 2).


Example 4

Solve for x in the equation x^2 - 8 = 0.

Solution:

Source: http://quicklatex.com/cache3/11/ql_ba1b4bc21549df33d84bd17dd3e41711_l3.png

The solution for x would be \pm \sqrt{8}. Note that \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}. Thus, the solution can also be expressed as \pm \, 2\sqrt{2}. Remember also that \sqrt{2} for example can be expressed as 2^{1/2}.


Example 5 (“Complex” Case)

Solve for x in the equation x^2 = -1.

Solution:

We have

Source: http://quicklatex.com/cache3/57/ql_f25e56f564aa3fc91b0bdc9bf469a957_l3.png

Here we do not have a difference of squares situation. The expression x^2 + 1 cannot be factored. If we were to take the square root of -1, it would not exist in the real numbers.

However, if we use complex/imaginary numbers where i = \sqrt{-1} then we can factor. Then we have x^2 + 1 = (x + i)(x - i). The imaginary numbered solutions to x^2 = -1 are \pm \sqrt{-1} = \pm i.


Notes

  • It is important to note that (a^{2} - b^2) is not the same as (b^2 - a^2). This is because order matters and factoring a (-1) from (b^2 - a^2) gives - \, (a^2 - b^2).
  • If you are new to factoring difference of squares, practice is recommended. Know your square numbers such as 1, 4, 9 ,16 , 25, 36 ,49,64, 81, 100, 121, 144, and 169 really well.

The featured math image was taken http://www.topuniversities.com/student-info/careers-advice/what-can-you-do-mathematics-degree.

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