If `alpha` and `beta` are zeroes of `x^(2)-5x+6`, then find the value of `alpha^(2)+beta^(2)`.

Explanation

To solve the problem, we need to find the value of \( \alpha^2 + \beta^2 \) given that \( \alpha \) and \( \beta \) are the roots (zeroes) of the polynomial \( x^2 - 5x + 6 \).

### Step-by-step Solution:

1. **Identify the polynomial**: The given polynomial is \( x^2 - 5x + 6 \).

2. **Use Vieta's formulas**: According to Vieta's formulas, for a quadratic polynomial of the form \( ax^2 + bx + c \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

Here, \( a = 1 \), \( b = -5 \), and \( c = 6 \).

3. **Calculate the sum of the roots**:
\[ \alpha + \beta = -\frac{-5}{1} = \frac{5}{1} = 5 \]

4. **Calculate the product of the roots**:
\[ \alpha \beta = \frac{6}{1} = 6 \]

5. **Use the identity for the sum of squares**: We know that:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]

6. **Substitute the values**:
- We have \( \alpha + \beta = 5 \) and \( \alpha \beta = 6 \).
\[ \alpha^2 + \beta^2 = (5)^2 - 2 \times 6 \]

7. **Calculate**:
\[ \alpha^2 + \beta^2 = 25 - 12 = 13 \]

8. **Final answer**:
\[ \alpha^2 + \beta^2 = 13 \]