Problem 14 - Precalculus Student Solutions Manual 5th

Question

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.

Step-by-Step Solution

Answer

The integers are 7 and 9.

Step 1: Define the variables

Let the first odd integer be represented by \(x\). The next consecutive odd integer is represented by \(x+2\).

Step 2: Write the difference of the squares

Use the formula for the difference of the squares of two consecutive odd integers: \[ (x+2)^2 - x^2 = 32 \]

Step 3: Expand the squares

Expand and simplify \[ (x+2)^2 - x^2 = x^2 + 4x + 4 - x^2 = 32 \]

Step 4: Simplify the equation

Combine like terms to simplify the equation: \[ 4x + 4 = 32 \]

Step 5: Solve for x

Isolate \(x\) by subtracting 4 from both sides: \[ 4x = 28 \] Divide by 4 to find \(x\): \[ x = 7 \]

Step 6: Find the consecutive odd integers

Since \(x = 7\), the first odd integer is 7. The next consecutive odd integer is \(x + 2 = 9\). Thus, the two consecutive odd integers are 7 and 9.

Key Concepts

Difference of Squares

Understanding the concept of the difference of squares can make solving many algebraic problems much simpler. The difference of squares is a specific mathematical formula:

\[ a^2 - b^2 = (a - b)(a + b) \]

This means that the difference between two squares can be factored into a product of the sum and the difference of their roots. For example, given the first odd integer as \(x\) and the next consecutive odd integer as \(x+2\), their squares would be \((x+2)^2\) and \(x^2\). The difference of their squares is then:

\[ (x+2)^2 - x^2 \]

This simplifies to the quadratic expression. Understanding this will help you efficiently solve problems by simplifying them through factorization. Seeing how the squares of integers relate can also make equations easier to digest and solve.

Odd Integers

Odd integers are key players in this exercise. An odd integer is an integer which is not divisible by 2. It will always have a remainder of 1 when divided by 2. Examples of odd integers include 1, 3, 5, 7, and so on.

In the exercise, we consider consecutive odd integers. If one odd integer is represented by \(x\), the next consecutive odd integer is represented by \(x+2\). For instance, if \(x=3\), then the next consecutive odd integer would be \(3+2=5\). This direct relationship makes it easier to handle problems where sequences of odd integers are involved. Understanding this relationship is fundamental in algebra and helps simplify the process of working with sequences of numbers.

Integer Equations

Solving integer equations can sometimes seem daunting, but breaking down the process step-by-step can make it manageable. The goal is to isolate the variable and find the integer that satisfies the equation.

Let's recap the steps from the solution provided:

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• Define the variables: Start by assigning variables to the integers.
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• Write the equation: Based on the problem statement, formulate the equation.
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• Expand the squares: Simplify any terms inside the equation.
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• Simplify the equation: Combine like terms to streamline the equation.
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• Solve for the variable: Isolate the variable and solve for it.

Using the problem as an example, we defined the first odd integer as \(x\) and wrote the difference of squares as \((x+2)^2 - x^2 = 32\). By solving this step by step, we simplified the equation and found that \(x=7\), leading us to identify the consecutive odd integers as 7 and 9. This structured approach can be applied to other integer equations, making them easier to solve effectively.