Question

If$\alpha$ and$\beta$ are the roots of the equation$2x(2x+1)=1$, $\beta$ then is equal to:

• A. $2\alpha (\alpha -1)$
• B. $-2\alpha (\alpha +1)$
• C. $2{\alpha }^2$
• D. $2\alpha (\alpha +1)$

Answer 1

The correct option is B

$-2\alpha (\alpha +1)$

Explanation for correct option:

Step 1: Find the sum and product of the roots of the given equation.

We have given an equation$2x(2x+1)=1$, whose roots are$\alpha$ and$\beta$

after simplifying the given equation, we get$4x^2+2x-1=0$

$\alpha$and$\beta$ are the roots of the above equation,

so, $\alpha +\beta =\frac{-b}{a}$

$\Rightarrow \alpha +\beta =\frac{-2}{4}=\frac{-1}{2}......\left(1\right)$

Step 2: Find$\mathit{\beta }$, from the given information.

$\alpha$ and$\beta$ will satisfy the given equation,

so,

$$\begin{gathered} 2\alpha (2\alpha +1)=1 \\ \alpha (2\alpha +1)=\frac{1}{2}....\left(2\right) \end{gathered}$$

put the equation$\left(2\right)$ in$\left(1\right)$,

$$\begin{gathered} \alpha +\beta =-\alpha (2\alpha +1) \\ \beta =-2{\alpha }^2-\alpha -\alpha \\ \beta =-2\alpha (\alpha +1) \end{gathered}$$

Hence, the option$\left(B\right)$ is the correct option.