The harmonic mean of the roots of the equation $(5+\sqrt{2}) x^{2}-(4+\sqrt{5}) x+8+2 \sqrt{5}=0$ is

Question

The harmonic mean of the roots of the equation $(5+\sqrt{2}) x^{2}-(4+\sqrt{5}) x+8+2 \sqrt{5}=0$ is\

Professor PC Thomas Classes · Answer

(b) : Let $\alpha, \beta$ are the roots of the given quadratic equation. Therefore, $\alpha+\beta=\frac{4+\sqrt{5}}{5+\sqrt{2}}$ and $\alpha \beta=\frac{8+2 \sqrt{5}}{5+\sqrt{2}}$\ Let $H$ be the harmonic mean between $\alpha$ and $\beta$.\ Then $H=\frac{2 \alpha \beta}{\alpha+\beta}=\frac{16+4 \sqrt{5}}{4+\sqrt{5}}=\frac{4(4+\sqrt{5})}{(4+\sqrt{5})}=4$.\