What is the sum of two consecutive even numbers, the difference of whose square is 84?

Options

A. 38
B. 34
C. 42
D. 46

Answer

To solve the problem of finding the sum of two consecutive even numbers whose squares differ by 84, we can follow these steps:

### Step 1: Define the two consecutive even numbers
Let the first even number be \( x \). Since we are looking for consecutive even numbers, the next consecutive even number will be \( x + 2 \).

### Step 2: Set up the equation for the difference of squares
According to the problem, the difference of their squares is 84. We can express this mathematically as:
\[ (x + 2)^2 - x^2 = 84 \]

### Step 3: Expand the left side of the equation
Now, we will expand the left side:
\[ (x + 2)^2 = x^2 + 4x + 4 \]
So, substituting this back into our equation gives:
\[ (x^2 + 4x + 4) - x^2 = 84 \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ 4x + 4 = 84 \]

### Step 5: Solve for \( x \)
Next, we will isolate \( x \):
\[ 4x = 84 - 4 \]
\[ 4x = 80 \]
\[ x = \frac{80}{4} = 20 \]

### Step 6: Identify the two consecutive even numbers
Now that we have \( x = 20 \), the two consecutive even numbers are:
- First number: \( 20 \)
- Second number: \( 20 + 2 = 22 \)

### Step 7: Calculate the sum of the two numbers
Finally, we find the sum of these two numbers:
\[ 20 + 22 = 42 \]

### Final Answer
The sum of the two consecutive even numbers is \( 42 \).

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