Show that the roots of the equation $x^{2} - 2 p x + p^{2} - q^{2} + 2 q r - r^{2} = 0$ are rational.

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Solution

The given equation is $x^{2} - 2 p x + p^{2} - q^{2} + 2 q r - r^{2} = 0$
The roots of the given equation are rational only when the discriminant is a perfect square.
The discriminant of the given equation is $b^{2} - 4 a c = (2 p)^{2} - 4 (p^{2} - q^{2} + 2 q r - r^{2})$
$\Rightarrow b^{2} - 4 a c = 4 p^{2} - 4 p^{2} + 4 (q^{2} - 2 q r + r^{2})$
$\Rightarrow b^{2} - 4 a c = 4 (q - r)^{2} = (2 q - 2 r)^{2}$
Here we can see that $b^{2} - 4 a c$ is a perfect square.
Therefore the roots of given equation are rational.

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