Find the harmonic mean of the roots of the equation `(5+sqrt2)x^2-(4+sqrt5)x+(8+2sqrt5)=0`

Options

A. 2
B. 4
C. 7
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 can follow these steps:

### Step 1: Identify coefficients
The coefficients of the quadratic equation are:
- \( A = 5 + \sqrt{2} \)
- \( B = -(4 + \sqrt{5}) \)
- \( C = 8 + 2\sqrt{5} \)

### Step 2: Calculate the sum of the roots
The sum of the roots \( \alpha + \beta \) can be found using the formula:
\[ \alpha + \beta = -\frac{B}{A} \]
Substituting the values:
\[ \alpha + \beta = -\frac{-(4 + \sqrt{5})}{5 + \sqrt{2}} = \frac{4 + \sqrt{5}}{5 + \sqrt{2}} \]

### Step 3: Calculate the product of the roots
The product of the roots \( \alpha \beta \) can be calculated using the formula:
\[ \alpha \beta = \frac{C}{A} \]
Substituting the values:
\[ \alpha \beta = \frac{8 + 2\sqrt{5}}{5 + \sqrt{2}} \]

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

### Step 5: Simplify the expression
This simplifies to:
\[ H_m = \frac{2(8 + 2\sqrt{5})}{4 + \sqrt{5}} \]
Now, simplifying further:
\[ H_m = \frac{16 + 4\sqrt{5}}{4 + \sqrt{5}} \]

### Step 6: Factor out common terms
We can factor out 4 from the numerator:
\[ H_m = \frac{4(4 + \sqrt{5})}{4 + \sqrt{5}} \]
The \( 4 + \sqrt{5} \) terms cancel out:
\[ H_m = 4 \]

### Final Answer
Thus, the harmonic mean of the roots of the given quadratic equation is:
\[ \boxed{4} \]