Find two numbers such that their sum, their product and the differences of their squares are equal.

Answer

To find two numbers \( a \) and \( b \) such that their sum, product, and the difference of their squares are equal, we can follow these steps:

### Step 1: Set up the equations
Let the two numbers be \( a \) and \( b \). According to the problem, we have:
1. Their sum: \( a + b \)
2. Their product: \( ab \)
3. The difference of their squares: \( a^2 - b^2 \)

We need to set these three expressions equal to a constant \( k \):
\[ a + b = ab = a^2 - b^2 = k \]

### Step 2: Express the difference of squares
We can express the difference of squares using the identity:
\[ a^2 - b^2 = (a + b)(a - b) \]
Since \( a + b = k \), we can substitute this into the equation:
\[ a^2 - b^2 = k(a - b) \]

### Step 3: Set up the equations
From the equations \( a + b = k \) and \( ab = k \), we can rewrite \( a \) in terms of \( b \):
\[ a = k - b \]

### Step 4: Substitute into the product equation
Substituting \( a \) into the product equation:
\[ (k - b)b = k \]
Expanding this gives:
\[ kb - b^2 = k \]
Rearranging leads to:
\[ b^2 - kb + k = 0 \]

### Step 5: Solve the quadratic equation
We can use the quadratic formula to solve for \( b \):
\[ b = \frac{k \pm \sqrt{k^2 - 4k}}{2} \]

### Step 6: Find the corresponding \( a \)
Once we find \( b \), we can find \( a \) using:
\[ a = k - b \]

### Step 7: Analyze the solutions
We need to ensure that both \( a \) and \( b \) are real numbers. This requires that the discriminant \( k^2 - 4k \) is non-negative:
\[ k^2 - 4k \geq 0 \]
Factoring gives:
\[ k(k - 4) \geq 0 \]
This inequality holds for \( k \leq 0 \) or \( k \geq 4 \).

### Step 8: Choose a suitable \( k \)
Let's choose \( k = 4 \) (the smallest positive integer satisfying the inequality):
\[ b = \frac{4 \pm \sqrt{4^2 - 4 \cdot 4}}{2} = \frac{4 \pm 0}{2} = 2 \]
Thus, \( b = 2 \) and substituting back gives:
\[ a = 4 - 2 = 2 \]

### Conclusion
The two numbers are \( a = 2 \) and \( b = 2 \).