Quadra Trigno Equation

Given equation

$x^{2}$ +px+q

Roots of the equation= $\tan 15$ , $\tan 30$

We know that

$\tan 15$ =2- $\sqrt{3}$

$\tan 30$ = $\frac {1}{\sqrt{3}}$

In the given equation

Sum of the roots = -p = 2- $\sqrt{3}$ + $\frac {1}{\sqrt{3}}$

$\Rightarrow$ p = $\frac {2-2\sqrt{3}}{\sqrt{3}}$

Product of the roots = q=(2- $\sqrt{3}$ )( $\frac {1}{\sqrt{3}}$ )

$\Rightarrow$ q= $\frac {2-\sqrt{3}}{\sqrt{3}}$

Therefore 2+q-p = 2+ $\frac {2-\sqrt{3}}{\sqrt{3}}$ - $\frac {2-2\sqrt{3}}{\sqrt{3}}$

$\Rightarrow$ 2+ $\frac {\sqrt{3}}{\sqrt{3}}$

$\Rightarrow$ 2+1

$\Rightarrow$ $\boxed{3}$

The exact value of $\tan { 15 } =2-\sqrt { 3 }$ from the formula: $\tan { (\alpha -\beta) = } \frac { \tan { \alpha -\tan { \beta } } }{ 1+\tan { \alpha \tan { \beta } } }$ , where $15=45-30$

The exact value of $\tan { 30 } =\frac { \sqrt { 3 } }{ 3 }$

From Vieta's formulas, we know that:

$\tan { 30 } +\tan { 15 } =-p$

$\tan { 30 } \times \tan { 15 } =q$

So long story short, solve for $p$ and $q$

Quite an easy sum.So,I suppose no solution would be required for this one.