Question

Use a Special Factoring Formula to factor the expression

$x^2+12x+36$

Short answer

The solution$x^2+12x+36={\left(x+6\right)}^2$

Step-by-step solution

Step 1. Apply the concept of factoring trinomials.

To factor a trinomial of the form $x^2+bx+c;a\ne 0$, we note that

$x^2+bx+c=\left(x+r\right)\left(x+s\right)=x^2+\left(s+r\right)x+rs$

So, we need to choose numbers r and s so that$s+r=b$ and $rs=c$.

Step 2. Analyze the trinomial.

Given$x^2+12x+36$

This is of the form$x^2+bx+c$ where$b=12$and$c=36$

The factorization of the trinomial will be$\left(x+r\right)\left(x+s\right)=x^2+\left(s+r\right)x+rs$

So, we need to choose numbers r and s so that$s+r=b$ and $rs=c$.

Step 3. Find the integers to factor the trinomial.

We need to choose integers r and s so that$s+r=12$ and$rs=-36$.

By trial and error we find that the two integers are $r=6,s=6$.

$s+r=6+6=12$

$rs=6\cdot 6=36$

Step 4. Factor the trinomiala.

The factorization of the trinomial is

$\begin{array}{l}{x}^2+12x+36=\left(x+r\right)\left(x+s\right)\\ \Rightarrow x^2+12x+36=\left(x+6\right)\left(x+6\right)\\ \Rightarrow x^2+12x+36={\left(x+6\right)}^2\end{array}$