2007
Problem - 2657
What real value of $x$ satisfies $\sqrt{5x} - \sqrt{2x} = 5-2$?
The answer is $\boxed{7+2\sqrt{10}}$.
\begin{align}
\sqrt{5x} &= 3+\sqrt{2x}\\
5x &= 9 + 2x + 6\sqrt{2x}\\
3x-9&= 6\sqrt{2x}\\
x-3&=2\sqrt{2x}\\
x^2-6x+9&=8x\\
x^2 - 14 x + 9&=0\\
x&=7\pm 2\sqrt{10}
\end{align}
Because $x$ must be positive in order to make $\sqrt{5x}$ defined. Thus the final answer is $7+2\sqrt{10}$.