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

Options

A. `2`
B. `4`
C. `6`
D. `8`

Answer

To find 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 will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the 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
For a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots \((\alpha + \beta)\) and the product of the roots \((\alpha \beta)\) can be calculated using the formulas:
- Sum of the roots: \(\alpha + \beta = -\frac{b}{a}\)
- Product of the roots: \(\alpha \beta = \frac{c}{a}\)

#### Calculate the sum of the roots:
\[ \alpha + \beta = -\frac{-(4 + \sqrt{5})}{5 + \sqrt{2}} = \frac{4 + \sqrt{5}}{5 + \sqrt{2}} \]

#### Calculate the product of the roots:
\[ \alpha \beta = \frac{8 + 2\sqrt{5}}{5 + \sqrt{2}} \]

### Step 3: Calculate the harmonic mean
The harmonic mean \(H\) of two numbers \(\alpha\) and \(\beta\) is given by the formula:
\[ H = \frac{2\alpha\beta}{\alpha + \beta} \]

Substituting the values we calculated:
\[ H = \frac{2 \cdot \frac{8 + 2\sqrt{5}}{5 + \sqrt{2}}}{\frac{4 + \sqrt{5}}{5 + \sqrt{2}}} \]

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

### Step 5: Further simplify
Now we simplify:
\[ H = \frac{16 + 4\sqrt{5}}{4 + \sqrt{5}} \]

### Step 6: Divide both terms
To simplify further, we can divide both terms in the numerator by \(4 + \sqrt{5}\):
\[ H = 4 \quad \text{(after cancelling \(4 + \sqrt{5}\))} \]

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