Question

If $\alpha$ is a root of the equation $2x(2x+1)=1,$ then the other root is

• A. $-2\alpha (\alpha +1)$
• B. $3{\alpha }^3-4\alpha$
• C. $3\alpha -4{\alpha }^3$
• D. None of these

Answer 1

The correct option is D None of these

Roots of the given equation $4x^2+2x-1=0$ are;

$\Rightarrow \frac{\sqrt{5}-1}{4}and-\frac{(\sqrt{5}+1)}{4}$

Which are equivalent to cos72 and - cos36

Let $cos72=\alpha$

Now - cos36=cos(36+180)=cos216

$\Rightarrow cos216=cos(3\times 72)=4co{s}^{3}72-3cos72$

$\therefore otherroot=cos216=4{\alpha }^3-3\alpha$

Hence answer is $4{\alpha }^3-3\alpha$