Let's solve the problem step-by-step.

Problem Restatement

We want to find the sum of two consecutive even numbers such that the difference of their squares is 84.

Step 1: Define the variables

Let the first even number be xxx.
Since the numbers are consecutive even numbers, the next even number is x+2x+2x+2.

Step 2: Write the equation for the difference of their squares

The difference of their squares is given as 84:

(x+2)2−x2=84(x+2)^2-x^2=84(x+2)2−x2=84

Step 3: Expand and simplify

(x2+4x+4)−x2=84(x^2+4x+4)-x^2=84(x2+4x+4)−x2=84

4x+4=844x+4=844x+4=84

Step 4: Solve for xxx

4x=84−44x=84-44x=84−4

4x=804x=804x=80

x=20x=20x=20

Step 5: Find the two numbers and their sum

The two consecutive even numbers are:

20and2220\quad \text{and}\quad 2220and22

Their sum is:

20+22=4220+22=4220+22=42

Final answer:

The sum of the two consecutive even numbers is 42.