Express r in terms of p and q- if the roots of the equation x2 - px - q - 0 are - and the roots of the equationx2 - rx - s - 0 are -4-4

Express r in terms of p and q if the roots of the equation $x^{2} + p x + q = 0$ are $α, β$ and the roots of the equation
$x^{2} - r x + s = 0$ are $α^{4}, β^{4}$

Q. If

α

and

β

are the roots of

x^{2} + p x + q = 0

and

α^{4}, β^{4}

are the roots of

x^{2} - r x + s = 0,

then the equation

x^{2} - 4 q x + 2 q^{2} - r = 0,

has always two real roots.

Q. If

α

and

β

are roots of

x^{2} + p x + q = 0

, then value of

α^{4} + β^{4}

in terms of

p

and

q

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Express r in terms of p and q, if the roots of the equation x2+px+q=0 are α,β and the roots of the equation x2−rx+s=0 are α4,β4

The correct option is C (p2−2q)2 - 2q2 α+β=−p, αβ=q α4+β4=r =(α2+β2)2 - 2α2β2 =((α+β)2−2αβ)2−2α2β2 r=(p2−2q)2−2q2

Express $r$ in terms of $p$ and $q$ , if the roots of the equation $x^{2} + p x + q = 0$ are $α, β$ and the roots of the equation
$x^{2} - r x + s = 0$ are $α^{4}, β^{4}$

The correct option is C
$(p^{2} - 2 q)^{2}$ - $2 q^{2}$

$α + β = - p, α β = q$
$α^{4} + β^{4} = r$
$= (α^{2} + β^{2})^{2}$ - 2 $α^{2} β^{2}$
$= ((α + β)^{2} - 2 α β)^{2} - 2 α^{2} β^{2}$
$r = (p^{2} - 2 q)^{2} - 2 q^{2}$