If `a-b= 7 and ab= 18`, find `a +b`

Answer

To solve the equations \( a - b = 7 \) and \( ab = 18 \) to find \( a + b \), we can follow these steps:

### Step 1: Express \( a \) in terms of \( b \)
From the first equation \( a - b = 7 \), we can express \( a \) as:
\[ a = b + 7 \]

### Step 2: Substitute \( a \) in the second equation
Now, substitute \( a \) in the second equation \( ab = 18 \):
\[ (b + 7)b = 18 \]
This simplifies to:
\[ b^2 + 7b = 18 \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ b^2 + 7b - 18 = 0 \]

### Step 4: Factor the quadratic equation
Next, we need to factor the quadratic equation \( b^2 + 7b - 18 = 0 \). We look for two numbers that multiply to \(-18\) and add to \(7\). The numbers \(9\) and \(-2\) work:
\[ (b + 9)(b - 2) = 0 \]

### Step 5: Solve for \( b \)
Setting each factor to zero gives us:
\[ b + 9 = 0 \quad \text{or} \quad b - 2 = 0 \]
Thus, we find:
\[ b = -9 \quad \text{or} \quad b = 2 \]

### Step 6: Find corresponding values of \( a \)
Now, we can find the corresponding values of \( a \) using \( a = b + 7 \):
1. If \( b = -9 \):
\[ a = -9 + 7 = -2 \]
2. If \( b = 2 \):
\[ a = 2 + 7 = 9 \]

### Step 7: Calculate \( a + b \)
Now we calculate \( a + b \) for both pairs:
1. For \( (a, b) = (-2, -9) \):
\[ a + b = -2 - 9 = -11 \]
2. For \( (a, b) = (9, 2) \):
\[ a + b = 9 + 2 = 11 \]

### Final Result
Thus, the possible values for \( a + b \) are:
\[ \text{Either } -11 \text{ or } 11 \]