The sum of the solutions of | x-2|+ x( x-4)+2=0, (x0), is equal to:

Question

The sum of the solutions of $|\sqrt x-2|+\sqrt x(\sqrt x-4)+2=0$, $(x>0)$, is equal to:

Accepted Answer

Let $a=\sqrt x$: $|a-2|+a^2-4a+2=0 \implies |a-2|+(a-2)^2-2=0$. Let $y=|a-2|\ge0$: $y+y^2-2=0 \implies y=1$ (rejecting $y=-2$).

$|a-2|=1 \implies a=1,3 \implies x=1,9$.

Sum $=1+9=10.$

Solutions · 1