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Suppose $a, b, c$ are rational numbers such that $a + bc, b + ac$ and $a + b$ are non-zero, and satisfy $${1\over{a+bc}}+{1\over{b+ac}}={1\over{a+b}}$$ Prove that $\sqrt{(c − 3)(c + 1)}$ is rational.

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A good approach is to try to show that the radicand is a perfect square.

Since $a, b, c$ are rational, therefore $a+bc, b+ac$ and $a+b$ are rational.

Also the given relation is rational.

So try to show that $(c-3)(c+1)$ is a perfect square.

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    $\begingroup$ Thanks, that worked @easymathematics. I was able to get a quadratic equation in a to make it a perfect square $\endgroup$
    – Righter
    Mar 5, 2021 at 12:06

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