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(B) Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $\beta$.The harmonic mean $(HM)$ of two numbers $\alpha$ and $\beta$ i

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$4$

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(B) Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $\beta$.The harmonic mean $(HM)$ of two numbers $\alpha$ and $\beta$ is given by $HM = \frac{2\alpha\beta}{\alpha + \beta}$.From the given equation $(5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + (8 + 2\sqrt{5}) = 0$,we have:Sum of roots $\alpha + \beta = -\frac{b}{a} = \frac{4 + \sqrt{5}}{5 + \sqrt{2}}$.Product of roots $\alpha\beta = \frac{c}{a} = \frac{8 + 2\sqrt{5}}{5 + \sqrt{2}} = \frac{2(4 + \sqrt{5})}{5 + \sqrt{2}}$.Now,substitute these into the $HM$ formula:$HM = \frac{2 	imes \frac{2(4 + \sqrt{5})}{5 + \sqrt{2}}}{\frac{4 + \sqrt{5}}{5 + \sqrt{2}}}$.$HM = \frac{4(4 + \sqrt{5})}{5 + \sqrt{2}} 	imes \frac{5 + \sqrt{2}}{4 + \sqrt{5}}$.$HM = 4$.

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

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