If `a + b= 9 and ab = -22`, find : `a^(2)- b^(2)`

Answer

To solve the problem, we need to find the value of \( a^2 - b^2 \) given that \( a + b = 9 \) and \( ab = -22 \).

### Step-by-Step Solution:

1. **Write down the equations:**
We have two equations:
\[ a + b = 9 \quad \text{(1)} \]
\[ ab = -22 \quad \text{(2)} \]

2. **Express \( b \) in terms of \( a \):**
From equation (1), we can express \( b \):
\[ b = 9 - a \]

3. **Substitute \( b \) in equation (2):**
Substitute \( b \) in equation (2):
\[ a(9 - a) = -22 \]
Expanding this gives:
\[ 9a - a^2 = -22 \]

4. **Rearrange the equation:**
Rearranging the equation to standard quadratic form:
\[ a^2 - 9a - 22 = 0 \]

5. **Factor the quadratic equation:**
We need to factor the quadratic equation \( a^2 - 9a - 22 = 0 \). We look for two numbers that multiply to \(-22\) and add to \(-9\). The numbers are \( -11 \) and \( 2 \):
\[ (a - 11)(a + 2) = 0 \]

6. **Find the values of \( a \):**
Setting each factor to zero gives us:
\[ a - 11 = 0 \quad \Rightarrow \quad a = 11 \]
\[ a + 2 = 0 \quad \Rightarrow \quad a = -2 \]

7. **Find the corresponding values of \( b \):**
Using \( b = 9 - a \):
- If \( a = 11 \):
\[ b = 9 - 11 = -2 \]
- If \( a = -2 \):
\[ b = 9 - (-2) = 11 \]

8. **Calculate \( a^2 - b^2 \):**
We can use the identity \( a^2 - b^2 = (a + b)(a - b) \):
\[ a + b = 9 \]
To find \( a - b \):
- If \( a = 11 \) and \( b = -2 \):
\[ a - b = 11 - (-2) = 11 + 2 = 13 \]
- If \( a = -2 \) and \( b = 11 \):
\[ a - b = -2 - 11 = -13 \]

So, in both cases:
\[ a^2 - b^2 = (a + b)(a - b) = 9 \times 13 = 117 \]

### Final Answer:
\[ a^2 - b^2 = 117 \]