Q. 192 - Intermediate Algebra

Question

In the following exercises, factor completely using the difference of squares pattern, if possible.

$p^2+14p+49-q^2$

Step-by-Step Solution

Answer

The factored form of the given expression is $(p+7-q)(p+7+q)$.

Step 1. Given information.

The given expression is:
$p^2+14p+49-q^2$

Step 2. Determine the factored form.

The given expression can be written as:
$\begin{array}{rcl}{p}^2+14p+49-q^2& =& (p^2+2(p\left)\right(7)+7^2)-q^2\\ & =& {(p+7)}^2-q^2 \left[\because {(a+b)}^2=a^2+2ab+b^2\right]\\ & =& (p+7-q)(p+7+q) \left[\because {a^2-b}^2=(a-b)(a+b)\right]\end{array}$

Step 3. Conclusion.

The factored form of the given expression is $(p+7-q)(p+7+q)$.