The sum of two numbers, as well as, the difference of their squares is 9. Find the numbers.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let the two numbers be \( \alpha \) and \( \beta \).

### Step 2: Set Up the Equations
According to the problem:
1. The sum of the two numbers is 9:
\[ \alpha + \beta = 9 \quad \text{(Equation 1)} \]
2. The difference of their squares is also 9:
\[ \alpha^2 - \beta^2 = 9 \quad \text{(Equation 2)} \]

### Step 3: Use the Difference of Squares Identity
Recall the identity:
\[ A^2 - B^2 = (A + B)(A - B) \]
Applying this to Equation 2, we get:
\[ (\alpha + \beta)(\alpha - \beta) = 9 \]
Substituting Equation 1 into this gives:
\[ 9(\alpha - \beta) = 9 \]

### Step 4: Solve for \( \alpha - \beta \)
Dividing both sides by 9:
\[ \alpha - \beta = 1 \quad \text{(Equation 3)} \]

### Step 5: Add Equations 1 and 3
Now, we will add Equation 1 and Equation 3:
\[ (\alpha + \beta) + (\alpha - \beta) = 9 + 1 \]
This simplifies to:
\[ 2\alpha = 10 \]
Dividing by 2:
\[ \alpha = 5 \]

### Step 6: Find \( \beta \)
Now substitute \( \alpha \) back into Equation 1 to find \( \beta \):
\[ 5 + \beta = 9 \]
Subtracting 5 from both sides:
\[ \beta = 4 \]

### Conclusion
The two numbers are:
\[ \alpha = 5 \quad \text{and} \quad \beta = 4 \]

### Final Answer
The two numbers are 5 and 4.

---