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

By Vieta's formulas for the equation x^2 + px + q = 0, the sum of the roots is -p and the product of the roots is q. This gives p = -(tan 15 + tan 30) and q = tan 15 * tan 30. Substituting these into 2 + q - p results in 2 + tan 15 * tan 30 + tan 15 + tan 30, which factors as 2 + tan 15(1 + tan 30). Using the known value tan 15 = 2 - sqrt(3) and tan 30 = 1/sqrt(3), we get tan 15(1 + tan 30) = (2 - sqrt(3))(1 + 1/sqrt(3)) = 2 - sqrt(3) + 2/sqrt(3) - 1 = 1 - sqrt(3) + 2/sqrt(3). Adding 2 to this quantity gives the final value of 3 + 2/sqrt(3) - sqrt(3) = 3.

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