First slide

Theory of expressions

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

Easy

A

2
B

4
C

6
D

8

Solution

Given equation is $(5 + \sqrt{2}) x^{2} - (4 + \sqrt{5}) x + 8 + 2 \sqrt{5} = 0$
Let x₁ and x₂ are the roots of the equation
$\Rightarrow x_{1} + x_{2} = \frac{4 + \sqrt{5}}{5 + \sqrt{2}}$ ………………….(i)
and $x_{1} x_{2} = \frac{8 + 2 \sqrt{5}}{5 + \sqrt{2}} = \frac{2 (4 + \sqrt{5})}{5 + \sqrt{2}} = 2 (x_{1} + x_{2})$ …(ii)
$∴ Harmonic mean = \frac{2 x_{1} x_{2}}{x_{1} + x_{2}} = \frac{4 (x_{1} + x_{2})}{(x_{1} + x_{2})} = 4$
[ from Eq. (ii)]

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