If `alpha != beta` but, `alpha^(2) = 4alpha - 2 and beta^(2) = 4beta - 2` then the quadratic equation with roots `(alpha)/(beta) and (beta)/(alpha)` is

Options

A. `x^(2) - 4x + 2 = 0`
B. `x^(2) - 6x + 1= 0`
C. `x^(2) + 6x - 1 = 0`
D. `x^(2) + 4x - 2 = 0`

Answer

To solve the problem, we need to find the quadratic equation with roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\) given the conditions on \(\alpha\) and \(\beta\).

### Step 1: Analyze the given equations
We start with the equations:
\[ \alpha^2 = 4\alpha - 2 \]
\[ \beta^2 = 4\beta - 2 \]
We can rearrange these equations to standard quadratic form:
\[ \alpha^2 - 4\alpha + 2 = 0 \]
\[ \beta^2 - 4\beta + 2 = 0 \]

### Step 2: Find the sum and product of the roots
Using Vieta's formulas for the quadratic equations, we can find the sum and product of the roots for both \(\alpha\) and \(\beta\).

For \(\alpha\):
- Sum of roots, \(\alpha + \beta = 4\)
- Product of roots, \(\alpha \beta = 2\)

### Step 3: Calculate the sum of the new roots
The new roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\). The sum of these roots can be calculated as follows:
\[ \text{Sum of roots} = \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} \]

### Step 4: Find \(\alpha^2 + \beta^2\)
We can express \(\alpha^2 + \beta^2\) using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the known values:
\[ \alpha^2 + \beta^2 = (4)^2 - 2(2) = 16 - 4 = 12 \]

### Step 5: Substitute back to find the sum of roots
Now substituting back into the sum of roots formula:
\[ \text{Sum of roots} = \frac{12}{2} = 6 \]

### Step 6: Calculate the product of the new roots
The product of the new roots is:
\[ \text{Product of roots} = \frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha} = 1 \]

### Step 7: Form the quadratic equation
Using the sum and product of the roots, we can write the quadratic equation in the form:
\[ x^2 - (\text{Sum of roots})x + (\text{Product of roots}) = 0 \]
Substituting the values we found:
\[ x^2 - 6x + 1 = 0 \]

### Final Answer
The quadratic equation with roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\) is:
\[ \boxed{x^2 - 6x + 1 = 0} \]