Problem 13 - 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 even integers is \(84 .\) Find the integers.

Step-by-Step Solution

Answer

The integers are 20 and 22.

Step 1 - Define the Integers

Let the first even integer be represented by \( x \). Since they are consecutive even integers, the next even integer will be \( x + 2 \).

Step 2 - Express their Squares

The squares of the integers are \( x^2 \) and \( (x + 2)^2 \) respectively.

Step 3 - Write the Equation

According to the problem, the difference of their squares is 84. Therefore, \( (x + 2)^2 - x^2 = 84 \).

Step 4 - Simplify the Equation

Expand and simplify the equation:\[ (x + 2)^2 - x^2 = 84 \]\[ x^2 + 4x + 4 - x^2 = 84 \]\[ 4x + 4 = 84 \]

Step 5 - Solve for x

Isolate \( x \):\[ 4x + 4 = 84 \]\[ 4x = 80 \]\[ x = 20 \]

Step 6 - Identify both Integers

If \( x = 20 \), the first even integer is 20 and the next consecutive even integer is \( 20 + 2 \), which is 22.

Key Concepts

Integer Sequences

Integer sequences are fundamental in mathematics. They help in understanding patterns and positions within numbers.
Consecutive integers are numbers that follow each other in order without gaps. For example, 1, 2, 3 are consecutive integers.
In this exercise, we focus on consecutive even integers. If you have an even integer represented by x, the next even integer is described by x + 2.
This keeps both numbers even, maintaining their sequence. In our problem, x and x + 2 are consecutive even integers forming a simple sequence. These sequences help visualizing and solving problems, especially when setting up equations.

Difference of Squares

The difference of squares is a special algebraic concept. It states that for any two numbers a and b:ewline ewline (a^2 - b^2) = (a - b)(a + b).
For our problem, we let a = x + 2 and b = x, representing consecutive even integers.
According to our equation, ewline ewline (x + 2)^2 - (x)^2 = 84.
We use the difference of squares to simplify equations because it transforms complex expressions into simpler multiplications.
In our example, solving ewline ewline (x + 2)^2 - x^2 = 4x + 4 simplifies the equation and makes it easier to find the value of x.

Algebraic Equations

Algebraic equations form the backbone of solving integer problems. They are mathematical statements asserting the equality of two expressions.
Our equation begins with (x + 2)^2 - x^2 = 84.
Expanding and simplifying gives 4x + 4 = 84.
To solve for x, we must isolate x by following these steps: ewline

• Subtract 4 from both sides: ewline 4x + 4 - 4 = 84 - 4. ewline This simplifies to:
• 4x = 80.
• Divide each side by 4 to get x: ewline 4x / 4 = 80 / 4 ewline x = 20

Understanding algebraic equations enables solving real-life problems by breaking down complex problems into simple, manageable steps. Our solution demonstrates this clearly, as we find the first even integer is 20, and the next consecutive even integer is 20 + 2, which equals 22.