Q. 132E - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

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 $u=x-2$

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