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Transformation of Roots: Linear Combination of Roots

If α, β are t...

Question

If $α, β$ are the roots of $x^{2} + 7 x + 5 = 0$ , then the equation whose roots are $α - 1, β - 1$ is

A

x^{2} + 5 x - 1 = 0

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B

x^{2} + 5 x + 1 = 0

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C

x^{2} + 7 x + 13 = 0

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D

x^{2} + 9 x + 13 = 0

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Solution

The correct option is D $x^{2} + 9 x + 13 = 0$
$f (x) = x^{2} + 7 x + 5 = 0 \to roots are α, β$
The equation whose roots are $α - 1, β - 1$ is,
$f (x + 1) = 0$
$\Rightarrow (x + 1)^{2} + 7 (x + 1) + 5 = 0 \Rightarrow x^{2} + 2 x + 1 + 7 x + 7 + 5 = 0 \Rightarrow x^{2} + 9 x + 13 = 0$

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