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

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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 $α + k$ and $β + k$ , then,

$(x - (α + k)) (x - (β + k)) = 0$

$(x - α - k) (x - β - k) = 0$

$x^{2} - β x - k x - α x + α β + k α - k x + k β + k^{2} = 0$

$x^{2} - x (β + k + α + k) + α β + k α + k β + k^{2} = 0$

$x^{2} - x (β + α + 2 k) + α β + k (α + β) + k^{2} = 0$

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

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

$a x^{2} - x (2 a k - b) + c - k b + a k^{2} = 0$

Therefore, the required equation is $a x^{2} - x (2 a k - b) + c - k b + a k^{2} = 0$ .

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