The harmonic mean of the roots of the equation `(5+sqrt(2))x^2-(4+sqrt(5))x+8+2sqrt(5)=0` is a. `2` b. `4` c. `6` d. `8`

Explanation

To find the harmonic mean of the roots of the equation \( (5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + (8 + 2\sqrt{5}) = 0 \), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 5 + \sqrt{2} \)
- \( b = -(4 + \sqrt{5}) \)
- \( c = 8 + 2\sqrt{5} \)

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = \frac{4 + \sqrt{5}}{5 + \sqrt{2}} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{8 + 2\sqrt{5}}{5 + \sqrt{2}} \)

### Step 3: Calculate the harmonic mean
The formula for the harmonic mean \( H \) of the roots \( \alpha \) and \( \beta \) is given by:
\[ H = \frac{2 \alpha \beta}{\alpha + \beta} \]
Substituting the values we found:
\[ H = \frac{2 \cdot \frac{8 + 2\sqrt{5}}{5 + \sqrt{2}}}{\frac{4 + \sqrt{5}}{5 + \sqrt{2}}} \]

### Step 4: Simplify the expression
The \( (5 + \sqrt{2}) \) in the numerator and denominator cancels out:
\[ H = 2 \cdot \frac{8 + 2\sqrt{5}}{4 + \sqrt{5}} \]

### Step 5: Factor out common terms
We can factor out a 2 from the numerator:
\[ H = 2 \cdot \frac{2(4 + \sqrt{5})}{4 + \sqrt{5}} = 2 \cdot 2 = 4 \]

### Final Answer
Thus, the harmonic mean of the roots of the equation is \( \boxed{4} \).

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