if the roots of the eq x2+px +q=0are tan 30 and tan 15 then find the value of2=p-q is - Maths - Complex Numbers and Quadratic Equations

Question

if the roots of the eq x2+px +q=0are tan 30 and tan 15 then find
the value of2=p-q is

Aakash Sharma · Accepted Answer

Given: tan 30Â° and tan 15Â° are the roots of the equation x2 + px + q = 0 â‡’ tan 30Â° + tan 15Â° = â€“p and tan 30Â° Ã— tan 15Â° = q $Now \tan 30 ° = \frac{1}{\sqrt{3}}$ and tan 15Â° = tan (60Â° â€“ 45Â°) $= \frac{\tan 60 ° - \tan 45 °}{1 + \tan 60 ° \tan 45 °} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}$ $\Rightarrow p = - \frac{1}{\sqrt{3}} - \frac{\sqrt{3} - 1}{\sqrt{3} + 1} and q = \frac{\sqrt{3} - 1}{\sqrt{3} (\sqrt{3} + 1)} \Rightarrow 2 + p - q = 2 - (\frac{1}{\sqrt{3}} + \frac{\sqrt{3} - 1}{\sqrt{3} + 1} + \frac{\sqrt{3} - 1}{\sqrt{3} (\sqrt{3} + 1)}) = 2 - (\frac{\sqrt{3} + 1 + 3 - \sqrt{3} + \sqrt{3} - 1}{3 + \sqrt{3}}) = 2 - 1 = 1$

if the roots of the eq. x2+px +q=0are tan 30 and tan 15 then find the value of2=p-q is