Chapter 9.4, Problem 45PPS

To determine

Find the solution.

Answer to Problem 45PPS

  $a=-9$ or $a=1$

Explanation of Solution

Given Information: $a^2+8a+16=25$

Formula Used: $a^2+2ab+b^2={(a+b)}^2$ and $a^2-b^2=(a+b)(a-b)$

Calculation:

  $a^2+8a+16=25$

So, simplify

  $a^2+8a+16-25=0$

Hence,

  $a^2+8a-9=0$

Now, adding & subtracting the square of half the coefficient of a.

  $a^2+8a-9+{\left(\frac{8}{2}\right)}^2-{\left(\frac{8}{2}\right)}^2=0$

Now use formula, $a^2+2ab+b^2={(a+b)}^2$

  ${\left(a+4\right)}^2-9-{\left(4\right)}^2=0$

So,

  ${\left(a+4\right)}^2-9-16=0$

Hence,

  ${\left(a+4\right)}^2-25=0$

Now, square root of 25 is 5.

  ${\left(a+4\right)}^2-{\left(5\right)}^2=0$

Now use formula, $a^2-b^2=(a+b)(a-b)$

  $\left(a+4+5\right)\left(a+4-5\right)=0$

Hence,

  $\left(a+9\right)\left(a-1\right)=0$

Now,

  $\left(a+9\right)=0$ or $\left(a-1\right)=0$

Hence,

  $a=-9$ or $a=1$