If the roots of the equation $x^{2} + a x + b = 0$ are $c$ and $d$ , then one of the roots of equation $x^{2} + (2 c + a) x + c^{2} + a c + b = 0$ is

$c$

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$2 d$

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$2 c$

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$d - c$

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Solution

The correct option is D
$d - c$

Given equation: $x^{2} + a x + b = 0$

Roots are: $c$ and $d$

$c + d = - a$ and $c d = b$

$\Rightarrow c^{2} + a c + b = 0$ . . . $(1)$

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

Using equation $(1)$ ,

$\Rightarrow x^{2} + (2 c + a) x = 0$

$\Rightarrow x [x + (2 c + a)] = 0$

$\Rightarrow x = - 2 c - a = - 2 c + c + d = d - c$

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