First slide

Theory of equations

Question

If the roots of the equation ${ax}^{2} + bx + c = 0 be α and β,$ then the roots of the equation ${cx}^{2} + bx + a = 0$ are

Moderate

A

$- α, - β$
B

$α, \frac{1}{β}$
C

$\frac{1}{α}, \frac{1}{β}$
D

None of these

Solution

$α, β$ are roots of ${ax}^{2} + bx + c = 0$
Let the roots of ${cx}^{2} + bx + a = 0$ be $α^{'}, β^{'},$ then $α^{'} + β^{'} = - \frac{b}{c} and α^{'} β^{'} = \frac{a}{c}$
but $\frac{α + β}{αβ} = \frac{- b / a}{c / a} = \frac{- b}{c} \Rightarrow \frac{1}{α} + \frac{1}{β} = α^{1} + β^{1}$
Hence $α^{'} = \frac{1}{α} and β^{'} = \frac{1}{β}$

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