Chapter 9.5, Problem 22PPS

To determine

Find the solution.

Answer to Problem 22PPS

  $x=-3.3$ or $x=0.3$

Explanation of Solution

Given Information: $x^2+3x+21=22$

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

Calculation:

  $x^2+3x+21=22$

So, simplify

  $x^2+3x+21-22=0$

Now,

  $x^2+3x-1=0$

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

  $x^2+3x-1+{\left(\frac{3}{2}\right)}^2-{\left(\frac{3}{2}\right)}^2=0$

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

  ${\left(x+\frac{3}{2}\right)}^2-1-{\left(\frac{3}{2}\right)}^2=0$

So,

  ${\left(x+\frac{3}{2}\right)}^2-1-\frac{9}{4}=0$

So,

  ${\left(x+\frac{3}{2}\right)}^2-\left(\frac{4+9}{4}\right)=0$

Now,

  ${\left(x+\frac{3}{2}\right)}^2-\left(\frac{13}{4}\right)=0$

Now, square root of 13 is $\sqrt{13}$ and square root of 4 is 2

  ${\left(x+\frac{3}{2}\right)}^2-{\left(\frac{\sqrt{13}}{2}\right)}^2=0$

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

  $\left(x+\frac{3}{2}+\frac{\sqrt{13}}{2}\right)\left(x+\frac{3}{2}-\frac{\sqrt{13}}{2}\right)=0$

Now,

  $x=-\frac{3-\sqrt{13}}{2}$ or $x=-\frac{3+\sqrt{13}}{2}$

Hence,

  $x=-3.30$ or $x=0.30$