Question

For a >0 , solve for the x the equation
$2lo{g}_{x}a+lo{g}_{ax}a+3lo{g}_{a^{2}x}a=0$

Answer 1

We know that $p^2+q^2=(p-q{)}^2+2pq$...(1)
where $p=x$, $q=\frac{x}{x+1}$
$\therefore p-q=\frac{x^2}{x+1}=t,pq=\frac{x^2}{x+1}=t$...(2)
Hence the given equation by (1) and (2) is

$\therefore x=\frac{1\pm \sqrt{5}}{2}$