Question

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

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is C

3

We know tan(30 + 15) = $\frac{tan30+tan15}{1-tan30tan15}$. The value on the LHS is 1 and the RHS can be calculated becuase numerator is sum of the roots and denominator is related to the product of the roots.

tan 30, tan 15 are the roots of the equation $x^2+px+q=0$
$\Rightarrow tan30+tan15=-p$
tan 30 + tan 15 = q

tan (30 + 15) = $\frac{tan30+tan15}{1-tan30tan15}$
tan 45 = $\frac{-p}{1-q}$
$\Rightarrow$ 1-q = -p
$\Rightarrow$q-p = 1
$\Rightarrow$ 2+q-p = 2+1 = 3