Question

If $\alpha$ is a root of $4x^2+2x-1=0$ then root is:

• A. $4{\alpha }^3+3\alpha$
• B. $4{\alpha }^3-3\alpha$
• C. None of these
• D. $3{\alpha }^3-4\alpha$

Answer 1

The correct option is B

$4{\alpha }^3-3\alpha$

$\alpha$ is a root $\Rightarrow 4{\alpha }^2+2\alpha -1=0$

Let the other root be $\beta$.

Then the sum of roots will be $\alpha +\beta =-\frac{1}{2}$

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

Now, $4{\alpha }^2+2\alpha -1=0$

$\Rightarrow 4{\alpha }^2=1-2\alpha$

$\Rightarrow 4{\alpha }^3=\alpha (1-2\alpha )$

$\Rightarrow 4{\alpha }^3=\alpha -2{\alpha }^2$

$\Rightarrow 4{\alpha }^3=2\alpha -\frac{1}{2}$

Subtracting $3\alpha$ from both sides.

$\Rightarrow 4{\alpha }^3-3\alpha =-\alpha -\frac{1}{2}=\beta$

Hence, $4{\alpha }^3-3\alpha$ is other root.