Question

In the following exercises, factor using substitution${(x-2)}^2-3(x-2)-54$

Short answer

The factors are$\left(x-11\right)\left(x+4\right)$

Step-by-step solution

Step 1. Given Information

The trinomial is${(x-2)}^2-3(x-2)-54$

Find factors using substitution method.

Step 2. Find the term for substitution

Rewrite the trinomial to prepare for the substitution

${\left(x-2\right)}^2-3\left(x-2\right)-54$

Substitute$x-2=u$

So, trinomial becomes$u^2-3u-54$

Step 3. Factor the trinomial

Split the middle term of trinomial to find factors

$$\begin{gathered} u^2-3u-54 \\ {u}^2-9u+6u-54 \end{gathered}$$

Factor it

$$\begin{gathered} u\left(u-9\right)+6\left(u-9\right) \\ \left(u-9\right)\left(u+6\right) \end{gathered}$$

Step 4. Substitute back u

Substitute back $u=x-2$

$$\begin{gathered} \left(x-2-9\right)\left(x-2+6\right) \\ \left(x-11\right)\left(x+4\right) \end{gathered}$$