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 that their sum is 3 and the sum of their squares is 12.

### Step-by-Step Solution:

1. **Assign Variables:**
Let the two numbers be \( x \) and \( y \).

2. **Set Up the Equations:**
From the problem, we have the following equations:
\[ x + y = 3 \quad \text{(1)} \]
\[ x^2 + y^2 = 12 \quad \text{(2)} \]

3. **Use the Identity for the Sum of Squares:**
We can use the identity:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
Substituting equation (1) into this identity gives:
\[ x^2 + y^2 = (3)^2 - 2xy \]
This simplifies to:
\[ x^2 + y^2 = 9 - 2xy \quad \text{(3)} \]

4. **Set Up the Equation Using (2) and (3):**
Now we can set the right-hand sides of equations (2) and (3) equal to each other:
\[ 12 = 9 - 2xy \]

5. **Solve for \( xy \):**
Rearranging the equation gives:
\[ 2xy = 9 - 12 \]
\[ 2xy = -3 \]
Dividing both sides by 2:
\[ xy = -\frac{3}{2} \]

6. **Conclusion:**
Therefore, the product of the two numbers \( x \) and \( y \) is:
\[ xy = -\frac{3}{2} \]

### Final Answer:
The product of the two numbers is \( -\frac{3}{2} \).