If the sum of two numbers is 52 and their product is 672, then find the absolute difference between the numbers.

Options

A. 73
B. 4
C. 32
D. 6

Answer

To solve the problem of finding the absolute 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 \( x \) and \( y \).

2. **Set Up the Equations**:
According to the problem, we have two equations:
- The sum of the numbers:
\[ x + y = 52 \]
(Equation 1)
- The product of the numbers:
\[ x \cdot y = 672 \]
(Equation 2)

3. **Express One Variable in Terms of the Other**:
From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = 52 - y \]

4. **Substitute into the Product Equation**:
Substitute \( x \) in Equation 2:
\[ (52 - y) \cdot y = 672 \]
Expanding this gives:
\[ 52y - y^2 = 672 \]
Rearranging it results in:
\[ y^2 - 52y + 672 = 0 \]
(Equation 3)

5. **Solve the Quadratic Equation**:
We will use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -52, c = 672 \):
- Calculate the discriminant:
\[ b^2 - 4ac = (-52)^2 - 4 \cdot 1 \cdot 672 = 2704 - 2688 = 16 \]
- Now apply the quadratic formula:
\[ y = \frac{52 \pm \sqrt{16}}{2} \]
\[ y = \frac{52 \pm 4}{2} \]

6. **Find the Values of \( y \)**:
This gives us two possible values for \( y \):
- \( y = \frac{56}{2} = 28 \)
- \( y = \frac{48}{2} = 24 \)

7. **Find the Corresponding Values of \( x \)**:
Using \( y = 28 \):
\[ x = 52 - 28 = 24 \]
Using \( y = 24 \):
\[ x = 52 - 24 = 28 \]
Thus, the two numbers are \( 28 \) and \( 24 \).

8. **Calculate the Absolute Difference**:
The absolute difference between the two numbers is:
\[ |x - y| = |28 - 24| = 4 \]

### Final Answer:
The absolute difference between the two numbers is \( 4 \).

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