Question

How do I solve this equation?

Answer 1

$2x(4-x)^{-1/2}-3(4-x)^{1/2}=(4-x)^{-1/2}\bigg(2x-3(4-x)\bigg)$

We effectively rewrite the equation as

$\dfrac{5x-12}{(4-x)^{1/2}}=0$

In order for the LHS to be defined, we need to restrict $4-x>0$, or $x$. Now, the LHS will vanish when the numerator is 0, which happens for

$5x-12=0\implies5x=12\implies x=\dfrac{12}5$

This value is indeed smaller than 4, so the solution is $x=\dfrac{12}5$.