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 odd integers is 32. Find the integers.

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Let the first positive odd integer be represented by \(x\). Since \(x\) is an odd integer, the next consecutive odd integer can be represented as \(x + 2\).

Write an expression for the difference of the squares of these two integers: \((x + 2)^2 - x^2\).

Set up the equation based on the problem statement: \((x + 2)^2 - x^2 = 32\).

Expand the squares: \((x + 2)^2 = x^2 + 4x + 4\), so the equation becomes \((x^2 + 4x + 4) - x^2 = 32\).

Simplify the equation by canceling \(x^2\) terms and solve the resulting linear equation for \(x\) to find the first odd integer, then find the second by adding 2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Consecutive Odd Integers

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

Difference of Squares

The difference of squares formula states that a² - b² = (a - b)(a + b). This identity simplifies expressions involving the difference between two squared terms, making it easier to solve equations involving squares.

Formulating and Solving Equations

Translating word problems into algebraic equations is essential. Here, representing the consecutive odd integers as variables and using the difference of squares formula allows setting up an equation to solve for the unknown integers.