If $α, β$ are the roots of the equation $a x^{2} + b x + c = 0$ , then form an equation whose roots are:
$2 α, 2 β$

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Solution

Since $α$ and $β$ are the roots of the equation $a x^{2} + b x + c = 0$ , then,

$α + β = - \frac{b}{a}$

And,

$α β = \frac{c}{a}$

If the roots of any equation is $2 α$ and $2 β$ , then,

$(x - 2 α) (x - 2 β) = 0$

$x^{2} - 2 β x - 2 α x + 4 α β = 0$

$x^{2} - 2 x (α + β) + 4 α β = 0$

$x^{2} - 2 x (- \frac{b}{a}) + 4 (\frac{c}{a}) = 0$

$a x^{2} + 2 b x + 4 c = 0$

Therefore, the required equation is $a x^{2} + 2 b x + 4 c = 0$ .

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