If `alphaandbeta` are roots of the equation `x^(2)-5x+6=0`, then the value of `alpha^(2)-beta^(2)`

Options

A. 5
B. `-5`
C. `+-5`
D. `+-4`

Answer

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

### Step-by-Step Solution:

1. **Identify the coefficients of the quadratic equation**:
The given equation is \( x^2 - 5x + 6 = 0 \).
Here, \( a = 1 \), \( b = -5 \), and \( c = 6 \).

2. **Use Vieta's formulas to find the sum and product of the roots**:
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{-5}{1} = 5 \).
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{6}{1} = 6 \).

3. **Find the roots of the quadratic equation**:
We can factor the quadratic equation:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \]
Thus, the roots are \( \alpha = 2 \) and \( \beta = 3 \) (or vice versa).

4. **Calculate \( \alpha^2 - \beta^2 \)**:
We can use the identity \( \alpha^2 - \beta^2 = (\alpha + \beta)(\alpha - \beta) \).
- We already found \( \alpha + \beta = 5 \).
- Now, calculate \( \alpha - \beta \):
\[ \alpha - \beta = 2 - 3 = -1 \quad \text{(or } 3 - 2 = 1\text{)} \]

5. **Substitute the values into the identity**:
Now, substitute the values into the identity:
\[ \alpha^2 - \beta^2 = (\alpha + \beta)(\alpha - \beta) = 5 \times (-1) = -5 \]
Alternatively, if we take \( \alpha - \beta = 1 \):
\[ \alpha^2 - \beta^2 = 5 \times 1 = 5 \]

6. **Conclusion**:
Therefore, the value of \( \alpha^2 - \beta^2 \) can be either \( -5 \) or \( 5 \).