Prove the following
Question:

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

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

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

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

  4. (4) $2 \alpha^{2}$


Correct Option: , 2

Solution:

(2) Let $\alpha$ and $\beta$ be the roots of the given quadratic equation,

$2 x^{2} \cdot 2 x-1=0$…(i)

Then, $\alpha+\beta=-\frac{1}{2} \Rightarrow-1=2 \alpha+2 \beta$

and $4 \alpha^{2}+2 \alpha-1=0$  $[\because \alpha$ is root of eq. (i) $]$

$\Rightarrow 4 \alpha^{2}+2 \alpha+2 \alpha+2 \beta=0 \Rightarrow \beta=-2 \alpha(\alpha+1)$

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