Form the equation whose roots are the squares of the sum and of the difference of the roots of
$2 x^{2} + 2 (m + n) x + m^{2} + n^{2} = 0$

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Solution

Let the roots of equation $2 x^{2} + 2 (m + n) x + m^{2} + n^{2} = 0$ be $a, b$
We have $a + b = - (m + n)$ and $a b = \frac{m^{2} + n^{2}}{2}$
Let the roots of equation $x^{2} + p x + q = 0$ be $(a + b)^{2}, (a - b)^{2}$
Sum of roots is $- p = (a + b)^{2} + (a - b)^{2} = 2 (a^{2} + b^{2}) = 2 ((a + b)^{2} - 2 a b) = 2 ((m + n)^{2} - m^{2} - n^{2}) = 4 m n$
$\Rightarrow p = - 4 m n$
product of roots is $q = (a + b)^{2} \times (a - b)^{2} = (m + n)^{2} \times ((m + n)^{2} - 2 (m^{2} + n^{2})) = (m + n)^{2} (- (m - n)^{2}) = - (m^{2} - n^{2})^{2}$
Therefore the required equation is $x^{2} - 4 m n x - (m^{2} - n^{2})^{2} = 0$

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