The value of $(7+\sqrt{x}+\frac{1}{5-\sqrt{x}})$ is rational for only one positive integer $x$ that is not a perfect square. What is the value of $x$?
I tried $x=1$ and i get
$$7+1+\frac{1}{4}=\frac{31}{4}$$
which is rational so im guessing $x=1$ but does anyone see how to solve this problem in a more systematic way.
The $7$ is irrelevant to the question.
$$\sqrt{x}+\frac{1}{5-\sqrt{x}}=\frac{(25-x)\sqrt{x}}{25-x}+\frac{5+\sqrt{x}}{25-x}=\frac{(26-x)\sqrt{x}+5}{25-x}$$
Do you see a value for $x$ that makes something special happen?