First slide

Theory of equations

Question

If $α$ and $β$ are the roots of the equation $2 x (2 x + 1) = 1$ ,then $β$ is equal to

Moderate

A

$2 α (α + 1)$
B

$- 2 α (α + 1)$
C

$2 α (α - 1)$
D

$2 α^{2}$

Solution

The given quadratic equation
$2 x (2 x + 1) = 1$
$\Rightarrow 4 x^{2} + 2 x - 1 = 0 has roots α and β$
so, $α + β = - \frac{1}{2} \Rightarrow 2 α + 2 β = - 1$
and $4 α^{2} + 2 α - 1 = 0$
From Eqs. (i) and (ii), we get
$\begin{aligned} 4 α^{2} + 2 α + (2 α + 2 β) = 0 \\ \Rightarrow 2 α^{2} + 2 α + β = 0 \Rightarrow β = - 2 α (α + 1) \end{aligned}$

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