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

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$- 2 α (α + 1)$

$3 α^{3} - 4 α$

$4 α^{3} - 3 α$

none of these

answer is C.

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Detailed 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,

$\begin{array}{l} α + β = - \frac{1}{2} \\ \Rightarrow β = - \frac{1}{2} - α \\ \Rightarrow β = - \frac{1 + 2 α}{2} \\ \Rightarrow β = - \frac{4 α^{2} + 2 α + 2 α}{2} \\ \Rightarrow β = - 2 α (α + 1) \\ \Rightarrow β = - 2 α^{2} - 2 α \\ \Rightarrow β = - 2 α \times α - 2 α \\ \Rightarrow β = α (4 α^{2} - 1) - 2 α \\ \Rightarrow β = 4 α^{3} - 3 α . \end{array}$