How do you solve 5(sqrt x) - x =6?

Jul 20, 2015

Let $t = \sqrt{x}$, solve the resulting quadratic in $t$,

then derive $x = 4$ or $x = 9$.

Explanation:

Let $t = \sqrt{x}$.

Note $t \ge 0$ since sqrt denotes the non-negative square root.

Then $5 t - {t}^{2} = 6$

Add ${t}^{2} - 5 t$ to both sides to get:

$0 = {t}^{2} - 5 t + 6 = \left(t - 2\right) \left(t - 3\right)$

So $t = 2$ or $t = 3$. These both satisfy $t \ge 0$ so are valid solutions.

So $x = {2}^{2} = 4$ or $x = {3}^{2} = 9$