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

Answer

To find \( \alpha^2 + \beta^2 \) where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 - 5x + 6 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = -5 \)
- \( c = 6 \)

### Step 2: Calculate the sum and product of the roots
Using the formulas for the sum and product of the roots:
- The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-5}{1} = 5 \]

- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{6}{1} = 6 \]

### Step 3: Use the identity for \( \alpha^2 + \beta^2 \)
We can use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]

### Step 4: Substitute the values
Now, substitute the values we found into the identity:
\[ \alpha^2 + \beta^2 = (5)^2 - 2 \cdot 6 \]

### Step 5: Calculate
Calculating the right-hand side:
\[ \alpha^2 + \beta^2 = 25 - 12 = 13 \]

### Final Answer
Thus, the value of \( \alpha^2 + \beta^2 \) is:
\[ \boxed{13} \]