If `alpha and beta` are roots of the equation `x^(2)-2x+1=0`, then the value of `(alpha)/(beta)+(beta)/(alpha)` is

Options

A. 4
B. 1
C. 2
D. 0

Answer

To solve the problem, we need to find the value of \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) given that \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2 - 2x + 1 = 0\).

### Step 1: Identify the coefficients of the quadratic equation
The given quadratic equation is:
\[ x^2 - 2x + 1 = 0 \]
We can compare this with the standard form of a quadratic equation \(ax^2 + bx + c = 0\). Here, we have:
- \(a = 1\)
- \(b = -2\)
- \(c = 1\)

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \(\alpha + \beta\) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-2}{1} = 2 \]
- The product of the roots \(\alpha \beta\) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{1}{1} = 1 \]

### Step 3: Find \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\)
We can express \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) as:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} \]
Now, we need to calculate \(\alpha^2 + \beta^2\).

### Step 4: Calculate \(\alpha^2 + \beta^2\)
Using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we found:
\[ \alpha^2 + \beta^2 = (2)^2 - 2(1) = 4 - 2 = 2 \]

### Step 5: Substitute back to find \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\)
Now substituting \(\alpha^2 + \beta^2\) and \(\alpha \beta\) into our expression:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} = \frac{2}{1} = 2 \]

### Final Answer
Thus, the value of \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) is:
\[ \boxed{2} \]