If the roots of the equation $x^{2} + px + q = 0$ are $ an{30{^\circ}}$ and $ an{15{^\circ}}$ respectively, then the value of $2 + q - p$ is

Question

If the roots of the equation $x^{2} + px + q = 0$ are $	an{30{^\circ}}$ and $	an{15{^\circ}}$ respectively, then the value of $2 + q - p$ is

Dr. Abubacker · Accepted Answer

(a)
Since, $	an{30{^\circ}}$ and $	an{15{^\circ}}$ are the roots of equation
$x^{2} + px + q = 0$
$	herefore\ \ 	an{30{^\circ} + 	an{15{^\circ} = - p}}$
And $	an{30{^\circ}	an{15{^\circ} = q}}$
Now, $2 + q - p = 2 + 	an{30{^\circ} + 	an{15{^\circ} + \left( 	an{30{^\circ} + 	an{15{^\circ}}} ight)}}$
$= 2 + 	an{30{^\circ} + 	an{15{^\circ} + 1 - 	an{30{^\circ}	an{15{^\circ}}}}}$
$\left( \because	an{45{^\circ} = \frac{	an{30{^\circ} + 	an{15{^\circ}}}}{1 - 	an{30{^\circ}	an{15{^\circ}}}}} ight)$
$\Rightarrow \ \ 2 + q - p = 3$

Q. If the roots of the equation $x^{2} + px + q = 0$ are $\tan{30{^\circ}}$ and $\tan{15{^\circ}}$ respectively, then the value of $2 + q - p$ is

Explanation

Reviewed by Dr. Abubacker

Want more practice?