Solve this following

Question:

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

  1. $2 \alpha^{2}$

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

  3. $-2 \alpha(\alpha+1)$

  4. $2 \alpha(\alpha-1)$


Correct Option: , 3

Solution:

$\alpha$ and $\beta$ are the roots of the equation $4 x^{2}+2 x-1=0$.

$4 \alpha^{2}+2 \alpha=1 \Rightarrow \frac{1}{2}=2 \alpha^{2}+\alpha$  .........(1)

$\beta=\frac{-1}{2}-\alpha$

using equation (1)

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

$\beta=-2 \alpha^{2}-2 \alpha$

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

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