First slide

Theory of expressions

Question

If tan 25° and tan 20° are roots of the
quadratic equation $x^{2} + 2 p x + q = 0,$ then $2 p - q$ is equal to

Moderate

A

-2
B

-1
C

0
D

1

Solution

We have
$\tan 25^{\circ} + \tan 20^{\circ} = - 2 p$
and $\tan 25^{\circ} \tan 20^{\circ} = q$
Now, $1 = \tan 45^{\circ} = \tan (25^{\circ} + 20^{\circ})$
$\begin{array}{l} = \frac{\tan (25^{\circ}) + \tan (20^{\circ})}{1 - \tan (25^{\circ}) \tan (20^{\circ})} = \frac{- 2 p}{1 - q} \\ \Rightarrow 1 - q = - 2 p \Rightarrow 2 p - q = - 1 \end{array}$

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