How to solve this equation without drawing it's graph

Question

I'm curious to know how can I solve this equation without drawing it's graph . Can we ever prove that the answer must be integer using number theory or other mathematical methods? $$\sqrt{72-\frac{72}{x}}+\sqrt{x-\frac{72}{x}}=x$$ ( the answer is $9$ )

Answer 1

If you are okay with using numerical methods you can easily use the fixed point iteration method to approximate the answer. Let,

$$f(x)=\sqrt{72 - \frac{72}{x}} + \sqrt{x - \frac{72}{x}}$$

To have a real value for $x$ we should have,

$$\sqrt{72 - \frac{72}{x}}\geq 0\mbox{ and }\sqrt{x - \frac{72}{x}}\geq 0$$

Then,

$$x\geq 1 \mbox{ and }x^2\geq 72\Rightarrow x>8$$

$$\therefore x>8$$

Now choose as the initial point any number greater than $8$ and carry out the fixed point method. You will see that value of $x$ converges to $9$.