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Q.
The harmonic mean of the roots of the equation $\left(2+\sqrt{3}\right)x^{2}-\left(3+\sqrt{5}\right)x+\left(6+2\sqrt{5}\right)=0$ is
Complex Numbers and Quadratic Equations
Solution:
Given equation is
$\left(2+\sqrt{3}\right)x^{2}-\left(3+\sqrt{5}\right)x+\left(6+2\sqrt{5}\right)=0$
let $\alpha, \beta$ be its roots
$\therefore \alpha+\beta=\frac{3+\sqrt{5}}{2+\sqrt{3}} $ and $\alpha\beta=\frac{6+2\sqrt{5}}{2+\sqrt{3}}$
$\therefore $ H.M. of $\alpha $ and $\beta$ is given by
$H=\frac{2\alpha\beta}{\alpha+\beta}=\frac{2\left(6+2\sqrt{5}\right)}{2+\sqrt{3}}\times\frac{2+\sqrt{3}}{3+\sqrt{5}}=4\left(1\right)=4$