If the roots of the quadratic equation x² + px + q = 0 are tan30° and tan15°, respectively then the value of 2 + q − p is

Given equation x² + px + q = 0

$Given roots are \tan {30}^0,\tan {15}^0$
$\Rightarrow$tan 30° + tan 15° = − p
and tan 30° ⋅ tan 15° = q
$$\begin{gathered} Now \tan {45}^{\circ }=\frac{\tan {30}^{\circ }+\tan {15}^{\circ }}{1-\tan {30}^{\circ }\tan {15}^{\circ }}=\frac{-\mathrm{p}}{1-\mathrm{q}} \\ \Rightarrow 1=\frac{-\mathrm{p}}{1-\mathrm{q}} \end{gathered}$$
⇒ − p = 1 − q
⇒ q − p = 1 

∴ Now the value of 2 + q − p = 2+1=3.