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  4. If the roots of the quadratic equation x2+px+q=0 are tan 30° , and tan 15° respectively, then the value of 2 + q - p is :

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

AIEEEAIEEE 2008

Solution:

Since tan 30° and tan 15° are roots of equation
$x^{2}+px+q=0.$
$\therefore $ tan 30° + tan 15° = - p
$and \, $ tan 30° tan 15° = q
Therefore
$2+q-p=2+tan $ 30° tan 15°
+ (tan 30° + tan 15°)
= 2+ tan 30° tan 15°
+1 - tan 30° tan 15°
$\left(\because tan 45^{\circ}=\frac{tan 30^{\circ}+tan 15^{\circ}}{1-tan 30^{\circ} tan 15^{\circ}}\right)$
$\Rightarrow \, 2+q -p=3$