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Relation between Roots and Coefficients for Quadratic

If α , β be...

Question

If $α, β$ be the roots of $x^{2} - p x + q = 0$ and $α^{'}, β^{'}$ be those of $x^{2} - p^{'} x + q^{'} = 0$ , find the value of ${(α - α^{'})}^{2} + (β - α^{'})^{2} + {(α - β^{'})}^{2} + {(β - β^{'})}^{2} .$ .

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Solution

$α, β$ are the roots of $x^{2} - p x + q = 0 ⟹ α + β = p, α β = q$
$α^{'}, β^{'}$ are the roots of $x^{2} - p^{'} x + q^{'} = 0 ⟹ α^{'} + β^{'} = p^{'}, α^{'} β^{'} = q^{'}$
$(α - α^{'})^{2} + (β - α^{'})^{2} + (α - β^{'})^{2} + (β - β^{'})^{2} = 2 (α^{2} + β^{2}) + 2 (α^{' 2} + β^{' 2}) - 2 α (α^{'} + β^{'}) - 2 β (α^{'} + β^{'})$
$= 2 (p^{2} - 2 q) + 2 (p^{' 2} - 2 q^{'}) - 2 p p^{'} = 2 p^{2} + 2 p^{' 2} - 2 p p^{'} - 4 q - 4 q^{'}$

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