First slide

Theory of equations

Question

If $α, β$ the roots of the equation ${ax}^{2} + bx + c = 0$ , then the value of $\frac{({aα}^{2} + c)}{(aα + b)} + \frac{({aβ}^{2} + c)}{(aβ + b)}$ is

Moderate

A

$\frac{b (b^{2} - 2 ac)}{4 a}$
B

$\frac{b^{2} - 4 ac}{2 a}$
C

$\frac{b (b^{2} - 2 ac)}{a^{2} c}$
D

none of these

Solution

We have ${aα}^{2} + c = - bα or aα + b = - \frac{c}{α}$
Similarly, $aβ + b = - \frac{c}{β}$
Hence, the given expression is
$\frac{b}{c} (α^{2} + β^{2}) = \frac{b (b^{2} - 2 ac)}{a^{2} c}$

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