The harmonic mean of the roots of the equatio | vedclass

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Question

(C) Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $\beta$.The harmonic mean $H$ of the roots is given by the formula $H

Vedclass · Accepted Answer

$4$

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(C) Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $\beta$.The harmonic mean $H$ of the roots is given by the formula $H = \frac{2\alpha\beta}{\alpha + \beta}$.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 $\alpha\beta = \frac{c}{a} = \frac{8 + 2\sqrt{5}}{5 + \sqrt{2}}$Substituting these values into the formula for $H$:$H = 2 \cdot \frac{\frac{8 + 2\sqrt{5}}{5 + \sqrt{2}}}{\frac{4 + \sqrt{5}}{5 + \sqrt{2}}}$$H = 2 \cdot \frac{8 + 2\sqrt{5}}{4 + \sqrt{5}}$$H = 2 \cdot \frac{2(4 + \sqrt{5})}{4 + \sqrt{5}}$$H = 2 \cdot 2 = 4$

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

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