The Harmonic Mean (H.M.) of the roots of the equation (5 + \(\sqrt{2}\))x² - (4 + \(\sqrt{5}\))x + (8 + \(2\sqrt{5}\)) = 0 is -

Formula Used:

If there is a quadratic equation ax² + bx + c = 0

Then,

Sum of roots (α + β) = −b/a and,

Product of roots (αβ) = c/a

If quadratic equation that will have two roots, then the harmonic mean (HM) of two numbers α and β is given by,

HM = \(\frac{2\alpha \beta}{\alpha +\beta }\)

Application:

We have quadratic equation,

(5 + \(\sqrt{2}\))x2 - (4 + \(\sqrt{5}\))x + (8 + \(2\sqrt{5}\)) = 0

Hence,

a = (5 + \(\sqrt{2}\))

b = - (4 + \(\sqrt{5}\))

c =  (8 + \(2\sqrt{5}\))

Using above formula,

HM = \(\frac{2\alpha \beta}{\alpha +\beta }\) .... (1)

We know that,

(αβ) = c/a = \(\frac{8+2\sqrt 5}{5+\sqrt 2}\)

(α + β) = −b/a = \(\frac{4+\sqrt 5}{5+\sqrt 2}\)

Putting the values of sum and product in equation (1),

\(\rm HM = \frac{2 \left( \frac{8 + 2 \sqrt 5}{5 + \sqrt 2} \right)}{\frac{4+\sqrt 5}{5 + \sqrt 2}}= \frac{2 ( 8 + 2 \sqrt 5)}{4 + \sqrt 5} = 2 (2) = 4\)