The difference of the squares of two consecutive even numbers is 84. What is the sum of the two numbers

Options

A. (a) 40
B. (b) 42
C. (c) 45
D. (d) 48

Answer

To solve the problem, we need to find two consecutive even numbers whose difference of squares is 84 and then calculate their sum. Let's denote the first even number as \( x \). The next consecutive even number would then be \( x + 2 \).

### Step-by-Step Solution:

1. **Define the numbers**: Let the first even number be \( x \) and the second consecutive even number be \( x + 2 \).

2. **Set up the equation**: According to the problem, the difference of the squares of these two numbers is 84. We can write this as:
\[ (x + 2)^2 - x^2 = 84 \]

3. **Expand the squares**: Now, we will expand the left-hand side of the equation:
\[ (x^2 + 4x + 4) - x^2 = 84 \]

4. **Simplify the equation**: The \( x^2 \) terms cancel out:
\[ 4x + 4 = 84 \]

5. **Isolate \( x \)**: Subtract 4 from both sides:
\[ 4x = 80 \]
Now, divide both sides by 4:
\[ x = 20 \]

6. **Find the two consecutive even numbers**: The first even number is \( x = 20 \) and the second is \( x + 2 = 22 \).

7. **Calculate the sum**: Now, we add the two numbers together:
\[ 20 + 22 = 42 \]

### Final Answer:
The sum of the two consecutive even numbers is **42**.

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