If `alpha " and " beta` are the roots of the quadratic equation `x^2 -5x +6 =0` then find `alpha/beta + beta/alpha`

Answer

To solve the problem, we need to find the value of \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) where \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2 - 5x + 6 = 0\).

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the form \(Ax^2 + Bx + C = 0\). Here, \(A = 1\), \(B = -5\), and \(C = 6\).

2. **Use Vieta's Formulas**:
According to Vieta's formulas, for a quadratic equation \(Ax^2 + Bx + C = 0\):
- The sum of the roots \(\alpha + \beta = -\frac{B}{A}\)
- The product of the roots \(\alpha \beta = \frac{C}{A}\)

For our equation:
\[ \alpha + \beta = -\frac{-5}{1} = 5 \]
\[ \alpha \beta = \frac{6}{1} = 6 \]

3. **Find \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\)**:
We can rewrite \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) using the identity:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} \]

4. **Calculate \(\alpha^2 + \beta^2\)**:
We can use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we found:
\[ \alpha^2 + \beta^2 = (5)^2 - 2 \cdot 6 = 25 - 12 = 13 \]

5. **Substitute into the equation**:
Now substituting \(\alpha^2 + \beta^2\) and \(\alpha \beta\) back into the equation:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{13}{6} \]

### Final Answer:
Thus, the value of \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) is \(\frac{13}{6}\).

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