First slide

Theory of equations

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

Moderate

A

2
B

4
C

6
D

8

Solution

Given equation is
$(5 + \sqrt{2}) x^{2} - (4 + \sqrt{5}) x + 8 + 2 \sqrt{5} = 0 Hence x_{1} + x_{2} = \frac{4 + \sqrt{5}}{5 + \sqrt{2}}$
and $x_{1} x_{2} = \frac{8 + 2 \sqrt{5}}{5 + \sqrt{2}} = \frac{2 (4 + \sqrt{5})}{5 + \sqrt{2}} = 2 (x_{1} + x_{2})$
Harmonic mean $= \frac{2 x_{1} x_{2}}{x_{1} + x_{2}} = \frac{2 x_{1} x_{2}}{x_{1} x_{2} / 2} = 4 .$

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