# Solve the following equation. Check all proposed solutions.

Solve the following equation. Check all proposed solutions. Show all work in solving and checking, and state your final answer

my work

First let's add the fractions on the left. To add any fraction, you must put them over a common denominator, in this case $(x+2)(x^2-4)$

Multiply out the numerator and denominator then combine like terms, but after doing all that I got

$-16=x^3+4x^2-17x$

and I can't get value for $x$

• Did you forget to factorise $(x^2 - 4)$ ? – Shailesh Sep 25 '15 at 15:16
• I already did (x-2)(x+2) but still stuck – user155971 Sep 25 '15 at 15:17
• Notice that $x=\pm 2$ cannot be solutions since they make the denominator zero on the left hand side. Therefore, we can divide by $x-2$ and get the below answer. – Kevin Sheng Sep 25 '15 at 15:23

You forgot to factorize $(x^2 - 4) = (x + 2)(x -2)$ from the denominator. Then the LCM becomes $(x^2 - 4)$ in which case the numerator gives
$5(x - 2) - 4 = (x-3)(x + 2)$, so you will get $(x^2-6x+8) = 0$ which is $(x- 4)(x-2) = 0$. Looking at the equation $\color{red}{x = 4}$ is the only solution. This is because when you put $x = 2$, the original equation is undefined.