Question

Find all solution of the equation.
$\frac{(2|x|-3{)}^2-|x|-6}{4x+1}=0$, which belong to the domain of definition of the function $y=(2x+1)/(x^2-36)$

Answer 1

for $f\left(x\right)=\frac{(2x+1)}{(x^2-36)}$

$domainϵR-\{-6,6\}$

thus $f\left(x\right)=\frac{({2|x|-3)}^2-|x|-6}{4x+1}=0$

$({2|x|-3)}^2=|x|+6={4|x|}^2+9-13|x|+3=0$

let $|x|=z$

$4z^2-13z+3=0⟹{4z}^2-12z-z+3=0⟹(4z-1)(z-3)=0$

$z=1/4$ and $z=3$

so $|x|=1/4⟹x=\pm 1/4$

and $|x|=3⟹x=\pm 3$