If - - be the roots of x2-px-q-0 and - - be the roots of x2-p-x-q-0- then the value of - - - -2 - - - -2 - - - - -2 - - - - -2

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Relationship between Zeroes and Coefficients of a Polynomial

If α, β be ...

Question

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

2 {p^{2} - 2 q + p^{' 2} - 2 q^{'} - p p^{'}}

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2 {p^{2} - 2 q + p^{' 2} - 2 q^{'} - q q^{'}}

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2 {p^{2} - 2 q - p^{' 2} - 2 q^{'} - p p^{'}}

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2 {p^{2} - 2 q - p^{' 2} - 2 q^{'} - q q^{'}}

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Solution

The correct option is B $2 {p^{2} - 2 q + p^{' 2} - 2 q^{'} - p p^{'}}$

For $x^{2} - p x + q = 0$
$α + β = p; α β = q$
For $x^{2} - p x + q = 0$
$α + β; α β = q$
$(α - α)^{2} + (β - α)^{2} + (α - β)^{2} + (β - β)^{2}$
$= 2 (α^{2} + β^{2}) + 2 (α^{2} + β^{2}) - 2 (α α + β α + α β + β β)$
$= 2 [(α + β)^{2} - 2 α β] + 2 [(α + β)^{2} - 2 α β]$
$= 2 [α (α + β) + β (α + β)]$
$= 2 [p^{2} - 2 q] + 2 [p^{2} - 2 q] - 2 (α + β) (α + β)$
$= 2 (p^{2} - 2 q + p^{2} - 2 q - p p)$

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If α,β be the roots of x2−px+q=0 and α′,β′ be the roots of x2−p′x+q′=0, then the value of (α−α′)2+(β−α)2+(α−β′)2+(β−β′)2

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