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Common Roots

If α and ...

Question

If $α$ and $β$ are the of ${a x}^{2} + b x + c = 0$ , find the equation whose roots are $α + 2$ and $β + 2$ .

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Solution

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

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