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

Given equation is

$2x\left(2x + 1\right) = 1 ⇒ 4{x}^{2} + 2x - 1 = 0$ .........(1)

roots of equation \left(1\right)$</p><p>are$\alpha and $\beta$

$∴ \alpha + \beta = - \dfrac{1}{2}$

$⇒ \beta = - \dfrac{1}{2} - \alpha$ ........(2)

and

$4{\alpha}^{2} + 2\alpha - 1 = 0 ⇒ {\alpha}^{2} = \dfrac{1}{4} - \dfrac{\alpha}{2}$ .........(3)

now

$- 2\alpha\left(\alpha + 1\right) = - 2{\alpha}^{2} - 2a$

$= - 2\left(\dfrac{1}{4} - \dfrac{\alpha}{2}\right) - 2\alpha = - \dfrac{1}{2} - \alpha = \beta$