The harmonic mean of roots of the equation $(5+\sqrt{2})x^{2}-(4+\sqrt{5})x+(8+2\sqrt{5})=0$ is:

Question

Accepted Answer

To determine the harmonic mean of the roots of the quadratic equation $(5+\sqrt{2})x^{2}-(4+\sqrt{5})x+(8+2\sqrt{5})=0$, we need to use the properties of quadratic equations and the definition of the harmonic mean.

### Explanation:

1. **Understanding the Harmonic Mean:**
   The harmonic mean $H$ of two numbers $a$ and $b$ is given by:
   $$
   H = \frac{2ab}{a+b}
   $$
   For a quadratic equation $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$, the harmonic mean of the roots can also be expressed using the relationships:
   $$
   r_1 + r_2 = -\frac{b}{a} \quad \text{and} \quad r_1r_2 = \frac{c}{a}
   $$
   Thus, the harmonic mean of the roots is:
   $$
   H = \frac{2 \cdot \frac{c}{a}}{-\frac{b}{a}} = \frac{2c}{-b}
   $$

2. **Applying to the Given Equation:**
   For the equation $(5+\sqrt{2})x^{2}-(4+\sqrt{5})x+(8+2\sqrt{5})=0$, we identify:
   - $a = 5 + \sqrt{2}$
   - $b = -(4 + \sqrt{5})$
   - $c = 8 + 2\sqrt{5}$

Substituting these into the formula for the harmonic mean:
   $$
   H = \frac{2c}{-b} = \frac{2(8 + 2\sqrt{5})}{4 + \sqrt{5}}
   $$

3. **Simplifying the Expression:**
   $$
   H = \frac{16 + 4\sqrt{5}}{4 + \sqrt{5}}
   $$
   To simplify, multiply the numerator and the denominator by the conjugate of the denominator:
   $$
   H = \frac{(16 + 4\sqrt{5})(4 - \sqrt{5})}{(4 + \sqrt{5})(4 - \sqrt{5})}
   $$
   The denominator simplifies to:
   $$
   (4 + \sqrt{5})(4 - \sqrt{5}) = 16 - 5 = 11
   $$
   The numerator simplifies to:
   $$
   (16 + 4\sqrt{5})(4 - \sqrt{5}) = 64 - 16\sqrt{5} + 16\sqrt{5} - 20 = 44
   $$
   Thus:
   $$
   H = \frac{44}{11} = 4
   $$

4. **Conclusion:**
   The harmonic mean of the roots is indeed 4, which corresponds to Option 2.

### Why Other Options are Incorrect:

- **Option 1 (2):** The calculation clearly shows that the harmonic mean is 4, not 2.
- **Option 3 (6):** The derived harmonic mean does not match 6.
- **Option 4 (8):** The harmonic mean calculation results in 4, not 8.

Therefore, the correct answer is Option 2, as the harmonic mean of the roots of the given quadratic equation is indeed 4.