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If $p, q$ and $r$ are rational numbers, then the roots of the equation $x^{2}-2 p x+p^{2}-q^{2}+2 q r-r^{2}=0$ are
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rational
The given equation is $x^{2}-2 p x+p^{2}-q^{2}+2 q r-r^{2}=0$ Now, discriminant $=(-2 p)^{2}-4(1)\left(p^{2}-q^{2}+2 q r-r^{2}\right)$
$=4 p^{2}-4 p^{2}+4 q^{2}-8 q r+4 r^{2}$
$=(2 q-2 r)^{2}=4(q-r)^{2}$
which is always greater than zero. Therefore, the roots of given equation are rational
$=4 p^{2}-4 p^{2}+4 q^{2}-8 q r+4 r^{2}$
$=(2 q-2 r)^{2}=4(q-r)^{2}$
which is always greater than zero. Therefore, the roots of given equation are rational
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