Problem 22 - Elementary Algebra

Question

The sum of the squares of two consecutive positive integers is 181 . Find the integers.

Step-by-Step Solution

Answer

The integers are 9 and 10.

Step 1: Understand the Problem

We need to find two consecutive positive integers such that the sum of their squares is 181.

Step 2: Define Variables

Let the first integer be \( n \). Since they are consecutive integers, the second integer will be \( n + 1 \).

Step 3: Formulate the Equation

The sum of the squares of these integers is given by \( n^2 + (n+1)^2 = 181 \).

Step 4: Expand the Equation

Expand \( (n+1)^2 \) to obtain \( n^2 + 2n + 1 = 181 \). The equation now becomes \( n^2 + n^2 + 2n + 1 = 181 \) or \( 2n^2 + 2n + 1 = 181 \).

Step 5: Simplify the Equation

Simplify the equation to get \( 2n^2 + 2n + 1 = 181 \). Subtract 181 from both sides to obtain \( 2n^2 + 2n + 1 - 181 = 0 \), which simplifies to \( 2n^2 + 2n - 180 = 0 \).

Step 6: Factor the Quadratic Equation

Divide the equation by 2 to simplify: \( n^2 + n - 90 = 0 \). Factor the quadratic equation to find the integers: \((n - 9)(n + 10) = 0 \).

Step 7: Solve for n

Solve each factor: \(n - 9 = 0\) or \(n + 10 = 0\). This gives \(n = 9\) or \(n = -10\). Since we are looking for positive integers, select \(n = 9\).

Step 8: State the Consecutive Integers

Given \(n = 9\), the consecutive integers are \(9\) and \(10\).

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow in order, without any gaps or breaks. For example, if you start counting from 1, consecutive integers would be 1, 2, 3, and so on. When dealing with math problems involving consecutive integers, it generally means numbers that are one unit apart from each other on the number line, such as 10 and 11. In our current exercise, we're dealing with two consecutive integers, which are the numbers that follow each other directly.

• If the first integer is denoted by \( n \), the next consecutive integer will be \( n + 1 \).
• When problems specify consecutive positive integers, it implies that both numbers are greater than zero.

Understanding consecutive integers is crucial because it helps simplify problems by setting up an organized way to calculate unknown values. This systematic approach is readily applied in algebra for problem solving.

Problem Solving

Problem solving in mathematics involves a series of strategic steps to find solutions. With word problems like ours, it's essential to break down the information given into manageable parts.
1. **Understand the problem**: Identify what is being asked and note the information provided. Here, we needed to find two consecutive integers whose squares add up to 181.
2. **Define variables**: Set variables to unknown quantities, helping translate the word problem into an equation. We defined the first integer as \( n \) and the second as \( n + 1 \).
3. **Create an equation**: Formulate an algebraic equation that reflects the problem statement. We set up our equation as \( n^2 + (n+1)^2 = 181 \).
By following these steps, complex problems become easier to handle. Organizing the information for an efficient solution through equations is at the heart of mathematical problem-solving.

Factoring

Factoring is a process used in algebra to express a polynomial as a product of its factors. It's particularly useful for solving quadratic equations like our example from the exercise.
When given a quadratic equation in the form \( ax^2 + bx + c = 0 \), factoring involves finding two numbers that multiply to \( ac \) and add up to \( b \).

• In our exercise, factored form turned out to be \((n - 9)(n + 10) = 0\).
• This step allowed us to use the Zero Product Property—if a product of two numbers is zero, at least one of the numbers must be zero. Solving this, we found \( n = 9 \) or \( n = -10 \).
• Since we sought positive integers, we discarded \( n = -10 \).

Factoring is effective for breaking down polynomial expressions into simpler elements. This leads directly to the values that satisfy the original equation.