First slide

Theory of equations

Question

If $α$ and $β$ , $α$ and $γ$ , $α$ and $δ$ are the roots of the equations ${ax}^{2} + 2 bx + c = 0, 2 {bx}^{2} + cx + a = 0$ and ${cx}^{2} + ax + 2 b = 0$ respectively, where a, b and c are positive real numbers, then $α + α^{2} =$

Moderate

A

$- 1$
B

$0$
C

$abc$
D

$a + 2 b + c$

Solution

Here $α + β = - \frac{2 b}{a}$
$γ + α = - \frac{c}{2 b}, α + δ = - \frac{a}{c}$
and $αβ = \frac{c}{a}, αγ = \frac{a}{2 b}, αδ = \frac{2 b}{c}$
$\Rightarrow α + δ = - \frac{1}{αβ}, α^{2} β + αβδ = - 1 . . . (i)$
$\Rightarrow α + β = - \frac{1}{αγ}, α^{2} γ + αβγ = - 1 . . . (ii)$
$\Rightarrow α + γ = - \frac{1}{αδ}, α^{2} δ + αβγ = - 1 . . . (iii)$
Solve equations (i), (ii) and (iii), we get $α = - 1$

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