Question

The harmonic mean of the roots of the equation

$(5 + \sqrt{2})x^{2} - (4 + \sqrt{3})x + 8 + 2\sqrt{3} = 0$ is

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (B)

4

Answer 2

Let α and β be the roots of the given equation

∴$a + \beta = \frac{4 + \sqrt{3}}{5 + \sqrt{2}}$, $\alpha\beta = \frac{8 + 2\sqrt{3}}{5 + \sqrt{2}}$

Hence, required harmonic mean

$= \frac{2\alpha\beta}{\alpha + \beta} = \frac{2\left( \frac{8 + 2\sqrt{3}}{5 + \sqrt{2}} \right)}{\frac{4 + \sqrt{3}}{5 + \sqrt{2}}} = \frac{2(8 + 2\sqrt{3})}{4 + \sqrt{3}} = \frac{4(4 + \sqrt{3})}{4 + \sqrt{3}} = 4$