First slide

Theory of equations

Question

If $α$ is a root of the equation $2 x (2 x + 1) = 1,$ then the other root, is

Moderate

A

$3 α^{3} - 4 α$
B

$- 2 α (α + 1)$
C

$4 α^{3} - 3 α$
D

none of these

Solution

Let $a, β$ be the roots of the equation $2 x (2 x + 1) = 1$ . Then, $α + β = - \frac{1}{2}$ and $α β = - \frac{1}{4}$
Again a is a root of $2 x (2 x + 1) = 0$
$∴ 4 α^{2} + 2 α - 1 = 0 . . . (i)$
Now,
$\begin{array}{l} α + β = - \frac{1}{2} \\ \Rightarrow β = - \frac{1}{2} - α \\ \Rightarrow β = - \frac{1 + 2 α}{2} \\ \Rightarrow β = - \frac{4 α^{2} + 2 α + 2 α}{2} \\ \Rightarrow β = - 2 α (α + 1) \\ \Rightarrow β = - 2 α^{2} - 2 α \\ \Rightarrow β = - 2 α \times α - 2 α \\ \Rightarrow β = α (4 α^{2} - 1) - 2 α \\ \Rightarrow β = 4 α^{3} - 3 α . \end{array}$

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