Factoring Using Perfect Square Trinomials

Solve the following polynomial equations.

a) \(\begin{align*}x^2 + 12x + 36 = 0\end{align*}\)

Rewrite:

The equation is already in the correct form.

Factor:

Rewrite \(\begin{align*}x^2 + 12x + 36 = 0\end{align*}\) as \(\begin{align*}x^2 + 2(6x) + 6^2 = 0\end{align*}\). We notice that this is a perfect square trinomial and we can factor it as \(\begin{align*}(x + 6)^2\end{align*}\).

Set the factor equal to zero:

\(\begin{align*} x + 6 = 0 \end{align*}\)

Solve:

\(\begin{align*} \underline{\underline{x = -6}}\end{align*}\)

Check: Substitute each solution back into the original equation.

\(\begin{align*}& (-6)^2 + 12(-6) + 36 = && \text{Substitute in -6.}\\ & 36 + -72 + 36 = && \text{Simplify.}\\ & 72 + -72 = 0&& \text{Checks out.}\end{align*}\)

b) \(\begin{align*}x^2 - 24x = -144\end{align*}\)

Rewrite: \(\begin{align*}x^2 - 24x = -144\end{align*}\) is rewritten as \(\begin{align*}x^2 - 24x + 144 = 0\end{align*}\)

Factor:

\(\begin{align*}x^2 - 24x + 144 =x^2+2(-12)x+(-12)^2=(x-12)^2\end{align*}\)

Set the factor equal to zero:

\(\begin{align*}x - 12 = 0\end{align*}\)

Solve:

\(\begin{align*}\underline{\underline{x = 12}} \end{align*}\)

Check: Substitute the solution back into the original equation.

\(\begin{align*}& (12)^2 - 24(12) + 144 = && \text{Substitute in 12.}\\ & 144 - 288+144 = && \text{Simplify.}\\ & 288- 288 = 0 && \text{Checks out.}\\ \end{align*}\)