If $α, β$ are the roots of the equation $a x^{2} + b x + c = 0$ , then the roots of the equation $a x^{2} + b x (x + 1) + c (x + 1)^{2} = 0$ are

α - 1, β - 1

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α + 1, β + 1

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\frac{α}{α - 1}, \frac{β}{β - 1}

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\frac{α}{1 - α}, \frac{β}{β - 1}

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Solution

The correct option is D $\frac{α}{1 - α}, \frac{β}{β - 1}$
Given: $α, β$ are the roots of the equation $a x^{2} + b x + c = 0$
To find the roots of the equation $a x^{2} + b x (x + 1) + c (x + 1)^{2} = 0$
Sol: According to the given criteria,
Sum of roots, $α + β = - \frac{b}{a} . . . . . . . . . . (i)$
Product of roots, $α β = \frac{c}{a} . . . . . . . . . . . . . . . (i i)$
Let $α_{1}, β_{1}$ be the roots of the new equation,
$a x^{2} + b x (x + 1) + c (x + 1)^{2} = 0$
Divide throughtout by $(x + 1)^{2}$ ,
$⟹ a . \frac{x^{2}}{(x + 1)^{2}} + b . \frac{x (x + 1)}{(x + 1)^{2}} + c . \frac{(x + 1)^{2}}{(x + 1)^{2}} = 0 ⟹ a . {(\frac{x}{x + 1})}^{2} + b . \frac{x}{x + 1} + c = 0$
Compare the above equation with $a x^{2} + b x + c = 0$
We get, $α = \frac{x}{x + 1}$
Hence the required roots will be, $x_{1} = \frac{α}{1 - α}, x_{2} = \frac{β}{β - 1}$

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