Let `alpha, beta` be the roots of the equation `x^2 +2 sqrt 2x-1 = 0`. The quadratic equation, whose roots are `alpha^(4) + beta^(4)` and `1/(10)(alpha^6 + beta^6)`, is:

Options

A. `x^2 - 180x + 9506 = 0`
B. `x^2 - 190x + 9466 = 0`
C. `x^2 - 195x + 9466 = 0`
D. `x^2 - 195x + 9506 = 0`

Answer

To solve the problem, we need to follow these steps:

### Step 1: Find the roots of the equation
The given quadratic equation is:
\[ x^2 + 2\sqrt{2}x - 1 = 0 \]

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 2\sqrt{2}, c = -1 \):
- Calculate the discriminant:
\[ b^2 - 4ac = (2\sqrt{2})^2 - 4(1)(-1) = 8 + 4 = 12 \]
- Now, substituting into the quadratic formula:
\[ x = \frac{-2\sqrt{2} \pm \sqrt{12}}{2} = \frac{-2\sqrt{2} \pm 2\sqrt{3}}{2} = -\sqrt{2} \pm \sqrt{3} \]

Thus, the roots are:
\[ \alpha = -\sqrt{2} + \sqrt{3}, \quad \beta = -\sqrt{2} - \sqrt{3} \]

### Step 2: Calculate \( \alpha^2 + \beta^2 \)
Using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
We know:
- \( \alpha + \beta = -2\sqrt{2} \)
- \( \alpha \beta = -1 \)

Now, substituting these values:
\[ \alpha^2 + \beta^2 = (-2\sqrt{2})^2 - 2(-1) = 8 + 2 = 10 \]

### Step 3: Calculate \( \alpha^4 + \beta^4 \)
Using the identity:
\[ \alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2(\alpha\beta)^2 \]
Substituting the values:
\[ \alpha^4 + \beta^4 = 10^2 - 2(-1)^2 = 100 - 2 = 98 \]

### Step 4: Calculate \( \alpha^6 + \beta^6 \)
Using the identity:
\[ \alpha^6 + \beta^6 = (\alpha^2 + \beta^2)(\alpha^4 + \beta^4) - \alpha^2\beta^2(\alpha^2 + \beta^2) \]
Substituting the known values:
- \( \alpha^2 + \beta^2 = 10 \)
- \( \alpha^4 + \beta^4 = 98 \)
- \( \alpha \beta = -1 \) implies \( \alpha^2 \beta^2 = 1 \)

Now substituting:
\[ \alpha^6 + \beta^6 = 10 \times 98 - 1 \times 10 = 980 - 10 = 970 \]

### Step 5: Find the second root \( \frac{1}{10}(\alpha^6 + \beta^6) \)
Calculating:
\[ \frac{1}{10}(\alpha^6 + \beta^6) = \frac{970}{10} = 97 \]

### Step 6: Form the quadratic equation
The roots of the new quadratic equation are \( \alpha^4 + \beta^4 = 98 \) and \( \frac{1}{10}(\alpha^6 + \beta^6) = 97 \).

The sum of the roots:
\[ 98 + 97 = 195 \]

The product of the roots:
\[ 98 \times 97 = 9506 \]

Thus, the quadratic equation is:
\[ x^2 - (sum \ of \ roots)x + (product \ of \ roots) = 0 \]
\[ x^2 - 195x + 9506 = 0 \]

### Final Answer
The quadratic equation is:
\[ x^2 - 195x + 9506 = 0 \]

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