Finding roots : $x-7\sqrt{x}+10$

Question

Find all possible roots of $k(x) = x-7\sqrt{x}+10$

I am having serious trouble rearranging this function as $ax^2+bx+c$ since it has $'-7\sqrt{x}'$. Can anyone please help me? Little help would really be appreciated.

Answer 1

WLOG $\sqrt x=y\implies x=y^2$

For real $y,x\ge0$

So we have $$y^2-7y+10=0$$

The problem could be more interesting had one of the roots been $<0$

Answer 2

$$ x-7\sqrt{x}+10=0 $$ $$ u^2 -7y+10=0 $$ $$ a=1, \quad b=-7, \quad c=10, \quad u = \frac{-b\pm\sqrt{b^2-4ac}}{2a}. $$ After you find $u,$ recall that $x=u^2.$

Answer 3

Factorize whenever possible: $x-7\sqrt{x}+10=(\sqrt x-2)(\sqrt x-5)$ from where the solutions follow immediately.