Chapter 9.4, Problem 8CYU

To determine

Find the solution.

Answer to Problem 8CYU

$m=-3$ or $m=-\frac{1}{2}$

Explanation of Solution

Given Information: $(4m+2)(3m+9)=0$

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

Calculation:

  $(4m+2)(3m+9)=0$

So,

  $12m^2+36m+6m+18\text{}=0$

Now, simplify

  $12m^2+42m+18\text{}=0$

Now, divide the equation by 6.

  $2m^2+7m+3\text{}=0$

Now, to make coefficient of $m^2$ = 1, divide the equation by 2.

  $m^2+\frac{7}{2}m+\frac{3}{2}\text{}=0$

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

  $m^2+\frac{7}{2}m+\frac{3}{2}\text{}+{\left(\frac{7}{4}\right)}^2-{\left(\frac{7}{4}\right)}^2=0$

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

  ${\left(m+\frac{7}{4}\right)}^2+\frac{3}{2}-{\left(\frac{7}{4}\right)}^2=0$

So,

  ${\left(m+\frac{7}{4}\right)}^2+\frac{3}{2}-\frac{49}{16}=0$

So,

  ${\left(m+\frac{7}{4}\right)}^2+\frac{24-49}{16}=0$

Hence,

  ${\left(m+\frac{7}{4}\right)}^2-\left(\frac{25}{16}\right)=0$

Now, square root of 25 is 5 and square root of 16 is 4.

  ${\left(m+\frac{7}{4}\right)}^2-{\left(\frac{5}{4}\right)}^2=0$

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

  $\left(m+\frac{7}{4}+\frac{5}{4}\right)\left(m+\frac{7}{4}-\frac{5}{4}\right)=0$

Now,

$\left(m+\frac{7}{4}+\frac{5}{4}\right)=0$ or $\left(m+\frac{7}{4}-\frac{5}{4}\right)=0$

So,

$\left(m+\frac{7+5}{4}\right)=0$ or $\left(m+\frac{7-5}{4}\right)=0$

So,

$\left(m+\frac{12}{4}\right)=0$ or $\left(m+\frac{2}{4}\right)=0$

Now,

$m=-\frac{12}{4}$ or $m=-\frac{2}{4}$

Hence,

$m=-3$ or $m=-\frac{1}{2}$