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Quadratic Equations

If α, β are...

Question

If $α, β$ are the roots of $x^{2} + a x + b = 0$ . Then prove that $\frac{α}{β}$ is a root of the equation $b x^{2} + (2 b - a^{2}) x + b = 0$ .

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Solution

Given $α, β$ are roots of the equation $x^{2} + a x + b = 0$ ,
Then, $α + β = - a, α β = b$ .......(1).
Now, we to prove $\frac{α}{β}$ to be the root of the equation $b x^{2} + (2 b - a^{2}) x + b = 0$ , we have to prove that
$b {(\frac{α}{β})}^{2} + (2 b - a^{2}) \frac{α}{β} + b = 0$
or $b (α^{2} + β^{2}) + (2 b - a^{2}) α β = 0$
Now,
$b (α^{2} + β^{2}) + (2 b - a^{2}) α β$
$= b {(α + β)^{2} - 2 α . β} + (2 b - a^{2}) α β$
$= b {(a)^{2} - 2 b} + (2 b - a^{2}) b$ [Using (1)]
$= b (a^{2} - 2 b) - (a^{2} - 2 b) b$
$= 0$

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