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Quantitative Aptitude

Maxima/Minima of a Quadratic Equation

If α, β are...

Question

If $α, β$ are the roots of the equation $x^{2} + p x + q = 0$ , find the value of $α^{3} β + α β^{3}$ .

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Solution

$α$ and $β$ one roots of $x^{2} + p x + q = 0$

$\Rightarrow α + β = - p$ and $α β = q$

We have to find $α^{3} β + α β^{3}$
$\Rightarrow (α β) (α^{2} + β^{3})$
$α^{2} + β^{2} = (α + β)^{2} - 2 α β$
$= (- p)^{2} - 2 (q) = p^{2} - 2 q$
$(α β) (α^{2} + β^{2}) = q (p^{2} - 2 q)$
$\Rightarrow p^{2} q - 2 q^{2}$
$α^{3} β + α β^{3} = p^{2} q - 2 q^{2}$

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If α,β are the roots of the equation x2+px+q=0, find the value of α3β+αβ3.

α and β one roots of x2+px+q=0⇒α+β=−p and αβ=qWe have to find α3β+αβ3 ⇒(αβ)(α2+β3)α2+β2=(α+β)2−2αβ =(−p)2−2(q)=p2−2q(αβ)(α2+β2)=q(p2−2q) ⇒p2q−2q2α3β+αβ3=p2q−2q2

If $α, β$ are the roots of the equation $x^{2} + p x + q = 0$ , find the value of $α^{3} β + α β^{3}$ .