If the harmonic mean of the roots of the equation $$(5 + \sqrt{2})x^{2} − bx + 8 + 2\sqrt{5} = 0$$ is 4 then the value of b is

Given quadratic equation is $$(5 + \sqrt{2})x^{2} − bx + 8 + 2\sqrt{5} = 0$$

Let the roots of the equation be $$\alpha\ $$ and $$\beta\ $$

So, harmonic mean of roots = $$\dfrac{2\alpha\ \beta\ }{\alpha\ +\beta\ }$$

Now, $$\alpha\cdot\beta\ $$ = Product of roots = $$\dfrac{\left(8+2\sqrt{\ 5}\right)}{\sqrt{\ 2}+5}$$

$$\alpha\ +\beta\ $$ = Sum of roots = $$-\dfrac{\left(-b\right)}{\sqrt{\ 2}+5}=\dfrac{b}{\sqrt{\ 2}+5}$$

So, Harmonic mean = $$2\cdot\dfrac{\dfrac{\left(8+2\sqrt{\ 5}\right)}{\sqrt{\ 2}+5}}{\dfrac{b}{\sqrt{\ 2}+5}}=4$$

or, $$2\cdot\dfrac{8+2\sqrt{\ 5}}{b}=4$$

or, $$2\cdot\dfrac{2\left(4+\sqrt{\ 5}\right)}{b}=4$$

or, $$b=4+\sqrt{\ 5}$$