Use the following facts. If x represents an integer, then x+1 represents the next consecutive integer. If x represents an even integer, then x+2 represents the next consecutive even integer. If x represents an odd integer, then x+2 represents the next consecutive odd integer. The difference of the squares of two positive consecutive even integers is 84. Find the integers.

Verified step by step guidance

1. Let the first even integer be represented by \(x\). Since \(x\) is an even integer, the next consecutive even integer can be represented as \(x + 2\).
2. Write an expression for the difference of the squares of these two integers: \( (x + 2)^2 - x^2 \).
3. Set up the equation based on the problem statement: \( (x + 2)^2 - x^2 = 84 \).
4. Expand the squares: \( (x^2 + 4x + 4) - x^2 = 84 \).
5. Simplify the equation and solve for \(x\): \$4x + 4 = 84\(, then isolate \)x$ to find the value of the first even integer.

Key Concepts

Consecutive Even Integers

Consecutive even integers are even numbers that follow one another in order, each differing by 2. For example, if x is an even integer, then x + 2 is the next consecutive even integer. Understanding this helps in setting up expressions for problems involving sequences of even numbers.

Difference of Squares

The difference of squares refers to the expression a² - b², which can be factored as (a - b)(a + b). This factorization simplifies solving equations involving the difference of two squared terms, making it easier to find unknown values.

Setting Up and Solving Algebraic Equations

Translating word problems into algebraic equations involves defining variables, expressing relationships, and solving for unknowns. In this problem, representing the integers with variables and using given conditions allows forming an equation to find the integers.