The difference between two positive numbers is 5 and the sum of their squares is 73. Find the product of these numbers.

Answer

To solve the problem step by step, we will denote the two positive numbers as \( x \) and \( y \).

### Step 1: Set up the equations
We know from the problem statement that:
1. The difference between the two numbers is 5:
\[ x - y = 5 \quad \text{(1)} \]
2. The sum of their squares is 73:
\[ x^2 + y^2 = 73 \quad \text{(2)} \]

### Step 2: Express one variable in terms of the other
From equation (1), we can express \( x \) in terms of \( y \):
\[ x = y + 5 \quad \text{(3)} \]

### Step 3: Substitute into the second equation
Now, substitute equation (3) into equation (2):
\[ (y + 5)^2 + y^2 = 73 \]

### Step 4: Expand the equation
Expanding \( (y + 5)^2 \):
\[ y^2 + 10y + 25 + y^2 = 73 \]
Combine like terms:
\[ 2y^2 + 10y + 25 = 73 \]

### Step 5: Rearrange the equation
Now, rearranging the equation gives:
\[ 2y^2 + 10y + 25 - 73 = 0 \]
\[ 2y^2 + 10y - 48 = 0 \]

### Step 6: Simplify the equation
We can divide the entire equation by 2:
\[ y^2 + 5y - 24 = 0 \]

### Step 7: Factor the quadratic equation
Next, we will factor the quadratic equation:
\[ (y + 8)(y - 3) = 0 \]
Setting each factor to zero gives:
\[ y + 8 = 0 \quad \Rightarrow \quad y = -8 \quad \text{(not valid since y is positive)} \]
\[ y - 3 = 0 \quad \Rightarrow \quad y = 3 \]

### Step 8: Find \( x \)
Now, substitute \( y = 3 \) back into equation (3) to find \( x \):
\[ x = y + 5 = 3 + 5 = 8 \]

### Step 9: Calculate the product
Now that we have both numbers, \( x = 8 \) and \( y = 3 \), we can find the product:
\[ x \cdot y = 8 \cdot 3 = 24 \]

### Final Answer
The product of the two numbers is:
\[ \boxed{24} \]

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