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 $9$ )
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1$\begingroup$ Hint : Take the square on both sides, isolate the square root, and take the square again. $\endgroup$– PeterDec 17, 2013 at 16:54
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$\begingroup$ @Peter not a fast method . I'm talking about guessing the answer set ( x is integer or not ) $\endgroup$– amatrDec 17, 2013 at 17:03
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1 Answer
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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 $9$.