If the sum of two numbers is 3 and the sum of their squares is 12, then their product is equal to :

Options

A. `3/2 `
B. `2/3`
C. `-3/2`
D. `-2/3`

Answer

To solve the problem, we need to find the product of two numbers given their sum and the sum of their squares. Let's break it down step by step.

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

### Step 2: Set Up the Equations
From the problem, we have the following equations:
1. The sum of the two numbers:
\[ x + y = 3 \]
2. The sum of their squares:
\[ x^2 + y^2 = 12 \]

### Step 3: Use the Identity for the Sum of Squares
We can use the identity that relates the sum of squares to the square of the sum:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
Substituting the values we have:
\[ x^2 + y^2 = (3)^2 - 2xy \]
This simplifies to:
\[ 12 = 9 - 2xy \]

### Step 4: Rearrange the Equation
Now, we will rearrange the equation to solve for \( xy \):
\[ 12 = 9 - 2xy \]
Subtract 9 from both sides:
\[ 12 - 9 = -2xy \]
\[ 3 = -2xy \]
Now, divide both sides by -2:
\[ xy = -\frac{3}{2} \]

### Step 5: Conclusion
Thus, the product of the two numbers \( x \) and \( y \) is:
\[ \boxed{-\frac{3}{2}} \]