First slide

Theory of equations

Question

If roots of the quadratic equation $a x^{2} + b x + c = 0$ are real and are of the form $\frac{α}{α - 1}, \frac{α + 1}{α}$ , then value of $(a + b + c)^{2}$ is

Moderate

A

$4 a c - b^{2}$
B

$b^{2} - 4 a c$
C

$c^{2} + a^{2} - 2 b^{2}$
D

none of these

Solution

$\frac{α}{α - 1} + \frac{α + 1}{α} = - \frac{b}{a}, and \frac{α}{α - 1} \cdot \frac{α + 1}{α} = \frac{c}{a}$
Now, $(a + b + c)^{2} = a^{2} {(1 + \frac{b}{a} + \frac{c}{a})}^{2}$
$= a^{2} {[1 - \{\frac{α}{α - 1} + \frac{α + 1}{α}\} + \frac{α + 1}{α - 1}]}^{2}$
$= a^{2} {(\frac{α}{α - 1} - \frac{α + 1}{α})}^{2}$
$= a^{2} [{(\frac{α}{α - 1} + \frac{α + 1}{α})}^{2} - 4 \frac{α}{α - 1} \cdot \frac{α + 1}{α}]$
$= a^{2} [\frac{b^{2}}{a^{2}} - \frac{4 c}{a}] = b^{2} - 4 a c$

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