Answer each question. Answer each question. Answer each question. Unknown NumbersUse 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 \( x \).
2. The next consecutive even integer can be represented by \( x + 2 \).
3. The square of the first integer is \( x^2 \) and the square of the second integer is \((x + 2)^2\).
4. According to the problem, the difference of their squares is 84, so we set up the equation: \((x + 2)^2 - x^2 = 84\).
5. Simplify and solve the equation: \((x^2 + 4x + 4) - x^2 = 84\) which simplifies to \(4x + 4 = 84\).

Key Concepts

Consecutive Even Integers

Consecutive even integers are pairs of even numbers that follow one another in the number line, differing by 2. For example, if x is an even integer, the next consecutive even integer can be expressed as x + 2. Understanding this concept is crucial for solving problems involving even integers, as it allows for the formulation of equations based on their properties.

Difference of Squares

The difference of squares is a mathematical identity that states a² - b² = (a - b)(a + b). This concept is particularly useful when dealing with problems involving the squares of numbers, as it simplifies the process of finding the difference between them. In the context of the question, it helps to express the relationship between the squares of two consecutive even integers.

Setting Up Equations

Setting up equations is a fundamental skill in algebra that involves translating a word problem or mathematical statement into a mathematical expression or equation. In this case, recognizing the relationship between the two consecutive even integers and their squares allows for the creation of an equation that can be solved to find the integers. This process is essential for effectively tackling algebraic problems.