Difference of Squares
\(a^2 - b^2 = (a + b)(a - b)\)
Example 1
Factorize \(400 - 9b^2\)
\begin{equation*}
\begin{split}
400 - 9b^2 & = 20^2 - (3b)^2\\\\
400 - 9b^2 & = (20 + 3b)(20 - 3b)
\end{split}
\end{equation*}
Example 2
Factorize \(x^8 - y^8\)
\begin{equation*}
\begin{split}
x^8 - y^8& = (x^4)^2 - (y^4)^2\\\\
x^8 - y^8& = (x^4 + y^4)(x^4 - y^4)\\\\
x^8 - y^8& = (x^4 + y^4)((x^2)^2 - (y^2)^2)\\\\
x^8 - y^8& = (x^4 + y^4)(x^2 + y^2)(x^2 - y^2)\\\\
x^8 - y^8& = (x^4 + y^4)(x^2 + y^2)(x + y)(x - y)
\end{split}
\end{equation*}
Example 3
Expand \((-8 + x)(-8 - x)\)
\begin{equation*}
\begin{split}
(-8 + x)(-8 - x) & = (-8)^2 - x^2\\\\
(-8 + x)(-8 - x) & = 64 - x^2
\end{split}
\end{equation*}
Example 4
Expand \((3x^2 - 2y^2)(3x^2 + 2y^2)\)
\begin{equation*}
\begin{split}
(3x^2 - 2y^2)(3x^2 + 2y^2) & = (3x^2)^2 - (2y^2)^2\\\\
(3x^2 - 2y^2)(3x^2 + 2y^2) & = 9x^4 - 4y^4
\end{split}
\end{equation*}