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:
\[\displaystyle (a + b)(a - b) = a^2 - ab + ab - b^2 = a^{2} - b^2 \]
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
\[x^2 - 144 = 0\]
\[(x + 12)(x - 12) = 0\]
\[x = \pm 12\]
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
\[x^4 - 16 = (x^2 + 4)(x^2 - 4) = (x^2 + 4)(x - 2)(x + 2)\]
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 (“Complex” Case)
Solve for \(x\) in the equation \(x^2 = -1\).
Solution
\[x^2 = -1\]
\[x = \pm \sqrt{-1}\]
\[x = \pm i\]
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 \(\pm \sqrt{-1} = \pm i\).
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.