91Ó°ÊÓ

The given equation is

$1-\frac{2}{m}=\frac{8}{m^2}$

Some terms of the equations are not defined for the $m=0$

so$1-\frac{2}{m}=\frac{8}{m^2},m≠0$

The least common denominator of all denominators in the equation is$m^2$

multiply both sides of the equation by the least common denominator $m^2$

localid="1647787450822" $$\begin{gathered} m^2\left(1-\frac{2}{m}\right)=m^2\left(\frac{8}{m^2}\right) \\ {m}^2\left(1\right)-m^2\left(\frac{2}{m}\right)=m^2\left(\frac{8}{m^2}\right) \\ {m}^2-2m=8 \end{gathered}$$

Factorize the equation

$$\begin{gathered} m^2-2m-8=0 \\ {m}^2-4m+2m-8=0 \\ m(m-4)+2(m-4)=0 \\ (m+2)(m-4)=0 \end{gathered}$$

Use the Zero Product Property

$$\begin{gathered} m+2=0 \\ m=-2 \end{gathered}$$

or$$\begin{gathered} m-4=0 \\ m=4 \end{gathered}$$

So solutions of the equation arerole="math" localid="1647787407698" $m=-2\&4$

Substitute $m=-2$in the equation

role="math" localid="1647787235545" $$\begin{gathered} 1-\frac{2}{m}=\frac{8}{m^2} \\ 1-\frac{2}{-2}\stackrel{?}{=}\frac{8}{{\left(-2\right)}^2} \\ 1+1\stackrel{?}{=}2 \\ 2=2 \end{gathered}$$

The left-hand side is equal to the right-hand side so the solution$m=-2$is correct.

Substitute $m=4$in the equation

$$\begin{gathered} 1-\frac{2}{m}=\frac{8}{m^2} \\ 1-\frac{2}{4}\stackrel{?}{=}\frac{8}{{\left(4\right)}^2} \\ \frac{4}{4}-\frac{2}{4}\stackrel{?}{=}\frac{8}{16} \\ \frac{2}{4}=\frac{2}{4} \end{gathered}$$

The left-hand side is equal to the right-hand side so the solution $x=4$is correct.