If the sum and the product of two numbers are 25 and 144 respectively then the difference of the numbers must be :

Options

A. 3
B. 5
C. 7
D. 11

Answer

To solve the problem of finding the difference between two numbers given their sum and product, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables**:
Let the two numbers be \( A \) and \( B \).
We know from the problem that:
\[ A + B = 25 \quad \text{(1)} \]
\[ A \times B = 144 \quad \text{(2)} \]

2. **Express A in terms of B**:
From equation (1), we can express \( A \) in terms of \( B \):
\[ A = 25 - B \quad \text{(3)} \]

3. **Substitute A in equation (2)**:
Substitute equation (3) into equation (2):
\[ (25 - B) \times B = 144 \]
Expanding this gives:
\[ 25B - B^2 = 144 \]
Rearranging it leads to:
\[ B^2 - 25B + 144 = 0 \quad \text{(4)} \]

4. **Solve the Quadratic Equation**:
We will use the quadratic formula to solve equation (4):
\[ B = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -25 \), and \( c = 144 \). Plugging in these values:
\[ B = \frac{25 \pm \sqrt{(-25)^2 - 4 \times 1 \times 144}}{2 \times 1} \]
\[ B = \frac{25 \pm \sqrt{625 - 576}}{2} \]
\[ B = \frac{25 \pm \sqrt{49}}{2} \]
\[ B = \frac{25 \pm 7}{2} \]

5. **Calculate the Values of B**:
This gives us two possible values for \( B \):
\[ B = \frac{32}{2} = 16 \quad \text{or} \quad B = \frac{18}{2} = 9 \]

6. **Find Corresponding Values of A**:
Using equation (3) to find \( A \):
- If \( B = 16 \):
\[ A = 25 - 16 = 9 \]
- If \( B = 9 \):
\[ A = 25 - 9 = 16 \]

7. **Calculate the Difference**:
Now we can find the difference \( |A - B| \):
\[ |A - B| = |16 - 9| = 7 \]

### Final Answer:
The difference between the two numbers is \( 7 \).

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