The harmonic mean of the roots of the equatio | vedclass

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(B) Given the quadratic equation $(5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0$.Let the roots be $x_1$ and $x_2$.From the properties of qua

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$4$

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(B) Given the quadratic equation $(5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0$.Let the roots be $x_1$ and $x_2$.From the properties of quadratic equations, the sum of roots is $x_1 + x_2 = \frac{4 + \sqrt{5}}{5 + \sqrt{2}}$.The product of roots is $x_1 x_2 = \frac{8 + 2\sqrt{5}}{5 + \sqrt{2}} = \frac{2(4 + \sqrt{5})}{5 + \sqrt{2}} = 2(x_1 + x_2)$.The harmonic mean $(HM)$ of two numbers $x_1$ and $x_2$ is given by $HM = \frac{2x_1 x_2}{x_1 + x_2}$.Substituting $x_1 x_2 = 2(x_1 + x_2)$ into the formula, we get $HM = \frac{2 \cdot 2(x_1 + x_2)}{x_1 + x_2} = 4$.

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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