The sum of two numbers is 7 and the sum their squares is 23, their product is equal to:

Options

A. 10
B. 11
C. 12
D. 13

Answer

To solve the problem step by step, we can follow these steps:

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

### Step 2: Set Up the Equations
From the problem, we know:
1. The sum of the two numbers:
\[ x + y = 7 \]
2. The sum of their squares:
\[ x^2 + y^2 = 23 \]

### Step 3: Use the Identity for the Sum of Squares
We can use the identity:
\[ (x + y)^2 = x^2 + y^2 + 2xy \]
Substituting the known values into this identity:
\[ (7)^2 = 23 + 2xy \]
This simplifies to:
\[ 49 = 23 + 2xy \]

### Step 4: Solve for \( 2xy \)
Rearranging the equation gives:
\[ 2xy = 49 - 23 \]
Calculating the right side:
\[ 2xy = 26 \]

### Step 5: Solve for \( xy \)
Dividing both sides by 2:
\[ xy = \frac{26}{2} = 13 \]

### Conclusion
The product of the two numbers \( x \) and \( y \) is:
\[ \boxed{13} \]