The sum of the squares of two numbers is 97 and the squares of their difference is 25. The product of the two numbers is

Options

A. 45
B. 36
C. 54
D. 63

Answer

To solve the problem step by step, let's define the two numbers as \( x \) and \( y \).

### Step 1: Write down the equations based on the problem statement.
According to the problem, we have two equations:
1. The sum of the squares of two numbers is 97:
\[ x^2 + y^2 = 97 \quad \text{(Equation 1)} \]
2. The square of their difference is 25:
\[ (x - y)^2 = 25 \quad \text{(Equation 2)} \]

### Step 2: Expand Equation 2.
We can expand Equation 2:
\[ (x - y)^2 = x^2 - 2xy + y^2 = 25 \]
Now we can substitute \( x^2 + y^2 \) from Equation 1 into this equation.

### Step 3: Substitute Equation 1 into the expanded Equation 2.
From Equation 1, we know that \( x^2 + y^2 = 97 \). So we can replace \( x^2 + y^2 \) in the expanded form:
\[ 97 - 2xy = 25 \]

### Step 4: Rearrange the equation to find \( xy \).
Now, let's isolate \( xy \):
\[ -2xy = 25 - 97 \]
\[ -2xy = -72 \]
Dividing both sides by -2 gives:
\[ xy = 36 \]

### Conclusion: The product of the two numbers.
Thus, the product of the two numbers \( x \) and \( y \) is:
\[ \boxed{36} \]