First slide

Theory of equations

Question

If $α, β$ are roots of the equation $x^{2} + 2 x + 4 = 0$ , then $\frac{1}{α^{3}} + \frac{1}{β^{3}}$ is equal to

Moderate

A

$- \frac{1}{2}$
B

$\frac{1}{2}$
C

$32$
D

$\frac{1}{4}$

Solution

Here, $α + β = - 2 and αβ = 4$
$∴ \frac{1}{α^{3}} + \frac{1}{β^{3}} = \frac{α^{3} + β^{3}}{{(αβ)}^{3}} = \frac{{(α + β)}^{3} - 3 αβ (α + β)}{{(αβ)}^{3}}$
$= \frac{{(- 2)}^{3} - 3 (- 2) (4)}{{(4)}^{3}} = \frac{16}{64} = \frac{1}{4}$

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