First slide

Theory of equations

Question

If $α, β$ are roots of the equation $a x^{2} + b x + c = 0$ then the equation whose roots are $2 α + 3 β$ and $3 α + 2 β$ is

Moderate

A

$a b x^{2} - (a + b) c x + (a + b)^{2} = 0$
B

$a c x^{2} - (a + c) b x + (a + c)^{2} = 0$
C

$a c x^{2} + (a + c) b x - (a + c) b x - (a + c)^{2} = 0$
D

none of these

Solution

We have,
$α + β = \frac{- b}{a}, α β = \frac{c}{a}$
The required equation is
$\begin{array}{l} x^{2} - 5 x (α + β) + (2 a + 3 β) (3 α + 2 β) = 0 \\ \Rightarrow x^{2} + \frac{5 x b}{a} + \{6 (α^{2} + β^{2}) + 13 α β\} = 0 \\ \Rightarrow x^{2} + \frac{5 b x}{a} + \{6 (α + β)^{2} + α β\} = 0 \\ \Rightarrow x^{2} + \frac{5 b}{a} x + (\frac{6 b^{2}}{a^{2}} + \frac{c}{a}) = 0 \\ \Rightarrow a^{2} x^{2} + 5 a b x + (6 b^{2} + a c) = 0 \end{array}$

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