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

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Solution

${a x}^{2} + b x + c = 0$ as roots $α & β$
sum of roots = $α + β = \frac{- b}{a}$
product of roots = $α β = \frac{c}{a}$
new roots are $α + 2 & β + 2$
sum = $α + 2 + β + 2 = \frac{- b}{a} + 4 = \frac{4 a - b}{a}$
product = $= (α + 2) (β + 2) = α β + 2 (α + β) + 4 = \frac{c}{a} - \frac{2 b}{a} + 4 = \frac{4 a - 2 b + c}{a}$
equation- $x^{2}$ -(sum of roots)x+product of roots=0
$x^{2} - (\frac{4 a - b}{a}) x + (\frac{4 a - 2 b + c}{a}) = 0 {a x}^{2} - (4 a - b) x + (4 a - 2 b + c) = 0$

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