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  4. If tan (π/12) is one of the roots of the equation 2 x2-b x+c=0,(b, c ∈ Q) then (b+c) equals

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Q. If $\tan \frac{\pi}{12}$ is one of the roots of the equation $2 x^2-b x+c=0,(b, c \in Q)$ then $(b+c)$ equals

Complex Numbers and Quadratic Equations

Solution:

$ \tan \frac{\pi}{12}=2-\sqrt{3}$, Conjugate $=2+\sqrt{3}$
$\Rightarrow$ Sum of roots $=\frac{b}{2}=2-\sqrt{3}+2+\sqrt{3}=4$
$\Rightarrow$ Product of roots $=\frac{ c }{2}=(2-\sqrt{3})(2+\sqrt{3})=1$
$\Rightarrow b = 8, c = 2$