First slide

Theory of equations

Question

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

Moderate

A

$a^{3} y^{2} + (b^{3} - 3 a b c) y + c^{3} = 0$
B

$a^{3} y^{2} + (3 a b c - b^{3}) y - c^{3} = 0$
C

$a^{2} y^{2} + 2 a b y + c^{2} = 0$
D

none of these

Solution

Put $y = α^{3} \Rightarrow α = y^{1 / 3}$
As $α$ is root of $a x^{2} + b x + c = 0$ , we get
$a y^{2 / 3} + b y^{1 / 3} + c = 0 \Rightarrow y^{1 / 3} (a y^{1 / 3} + b) = - c$
Cubing both the side, we get
$\begin{array}{l} y {(a y^{1 / 3} + b)}^{3} = - c^{3} \\ \Rightarrow y [a^{3} y + b^{3} + 3 a b y^{1 / 3} (a y^{1 / 3} + b)] = - c^{3} \end{array}$
$\Rightarrow y [a^{3} y + b^{3} + 3 a b (- c)] = - c^{3}$
$\Rightarrow a^{3} y^{2} + (b^{3} - 3 a b c) y + c^{3} = 0$

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