First slide

Theory of equations

Question

If $α$ and p are the roots of ax²+ bx² c = 0 then the equation ax² - bx(x - 1) + c(x -1)² = 0 has roots

Moderate

A

$\frac{α}{1 - α}, \frac{β}{1 - β}$
B

$\frac{1 - α}{α}, \frac{1 - β}{β}$
C

$\frac{α}{α + 1}, \frac{β}{β + 1}$
D

$\frac{α + 1}{α}, \frac{β + 1}{β}$

Solution

We have, ${ax}^{2} - bx (x - 1) + c (x - 1)^{2} = 0$ …… (i)
$\Rightarrow a {(\frac{x}{1 - x})}^{2} + b (\frac{x}{1 - x}) + c = 0$
Also , $α$ and $β$ be the roots of ${ax}^{2} + bx + c = 0$
$∴ α = \frac{x}{1 - x} and β = \frac{x}{1 - x} \Rightarrow x = \frac{α}{α + 1}, x = \frac{β}{β + 1}$
$Hence, \frac{α}{α + 1} and \frac{β}{β + 1}$ are the required roots.

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