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.
