Question

The difference of the squares of two positive consecutive even integers is 84. Find the integers. Use the fact​ that, if x represents an even​ integer, then x + 2 represents the next consecutive even integer.

Answer 1

The integers are 20 and 22

Explanation:

Let the first even integer be x . Then the next consecutive even integer
= x + 2
The relationship between these two integers is the difference of the squares = 84

This means
(x + 2)² - x² = 84

Use the relationship a² - b² = (a + b) (a -b)

Here a = x + 2 and b = x

So

a + b = (x + 2 ) + x = x + 2 + x = x + x + 2 = 2x + 2
a - b = (x + 2) - x = x + 2 - x = x -x + 2 = 2

Therefore

(a + b)(a - b) in terms of x is
(2x + 2) x 2 = 4x + 4

We know this is 84

So 4x + 4 = 84

Subtract 4 on both sides:
4x = 84 -4

4x = 80

x = 20

So the next consecutive even integer = 22

The integers are 20 and 22