Problem 23 - Elementary Algebra

Question

For two consecutive positive odd integers, the product of twice the smaller and the larger is \(126 .\) Find the integers.

Step-by-Step Solution

Answer

The integers are 7 and 9.

Step 1: Define the Variables

Let's define the smaller odd integer as \( x \). Since the integers are consecutive odd numbers, the next consecutive odd integer is \( x + 2 \).

Step 2: Set Up the Equation

According to the problem, the product of twice the smaller integer and the larger integer equals 126. This gives us the equation:\[ 2x(x + 2) = 126 \]

Step 3: Simplify the Equation

First, distribute the \( 2x \) to both terms inside the parentheses:\[ 2x^2 + 4x = 126 \]

Step 4: Rearrange Into a Standard Quadratic Equation

To rearrange, subtract 126 from both sides:\[ 2x^2 + 4x - 126 = 0 \]

Step 5: Simplify the Quadratic Equation

Divide every term by 2 to simplify:\[ x^2 + 2x - 63 = 0 \]

Step 6: Factor the Quadratic Equation

We need factors of \(-63\) that add up to \(2\). The factors \(9\) and \(-7\) work:\[ (x + 9)(x - 7) = 0 \]

Step 7: Solve for x

Set each factor equal to zero:\[ x + 9 = 0 \quad \text{or} \quad x - 7 = 0 \]Solving these, we get \( x = -9 \) or \( x = 7 \). Since we need positive integers, \( x = 7 \).

Step 8: Find the Consecutive Odd Integer

If \( x = 7 \), then the next consecutive odd integer is \( x + 2 = 9 \).

Key Concepts

Quadratic Equations

Quadratic equations are a vital part of algebra. They are polynomial equations of degree two, often written in the standard form:

• \(ax^2 + bx + c = 0\)

where \(a, b,\) and \(c\) are constants, and \(x\) represents an unknown variable.
When we talk about solving quadratic equations, we are looking for the values of \(x\) that make the expression equal to zero.
Quadratic equations can be solved through various methods, such as:

• Factoring
• Completing the square
• Using the quadratic formula: \(x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{2a}\)
• Graphing

In this context, the quadratic equation \(x^2 + 2x - 63 = 0\) is derived from translating the given word problem into a mathematical form.
The equation needs to be solved to find the unknown integer \(x\). These solutions are critical for finding the consecutive odd integers in the problem.

Factoring

Factoring is a common method used to solve quadratic equations. To factor a quadratic, we look for two numbers that multiply to give the constant term (

• \(c\):

in the equation) and add up to the linear coefficient (

• \(b\):

in the equation).
An example from our problem is factoring the equation \(x^2 + 2x - 63 = 0\). Here, we search for two numbers that multiply to \(-63\) and add to \(2\).
With trial and error or systematic thinking, we find the numbers 9 and -7, because:

• 9 imes -7 = -63
• 9 + (-7) = 2

The equation can be expressed as:

• \((x + 9)(x - 7) = 0\)

This method is helpful because it breaks down the equation into simpler parts, making it easier to find the roots or solutions.

Consecutive Integers

Consecutive integers are numbers that follow each other in order without any gaps. When we deal with odd integers, the sequence will consist of numbers with an increment of 2.

• Example: 1, 3, 5, etc.

In a problem involving consecutive integers, often only one starting number is involved, and the rest can be determined.
For our problem, the consecutive odd integers are noted as \(x\) and \(x + 2\). These numbers need to satisfy the conditions set by any given equation. When we discovered \(x = 7\) from solving the quadratic, it made finding the next number easy: \(x + 2 = 9\).
These steps help ensure we have the right integers to fit the original conditions described in a problem statement.

Mathematical Problem Solving

Mathematical problem solving is all about applying logic and known techniques to find solutions. Breaking down a complex problem into manageable parts is key.
Here is how a problem-solving workflow may look:

• Understand the problem: Make sure what needs to be solved is clear. Write down what is known and what needs to be found.
• Translate words to math: Convert the words in a problem into mathematical expressions or equations.
• Solve: Use appropriate mathematical methods, such as solving a quadratic equation by factoring.
• Check your work: Verify the solution meets the problem's original conditions.
• Draw insights: Consider what was learned and how it can apply to other problems.

In this series of steps, the required skills range from reading comprehension to equation solving. Effectively using them requires practice to develop these problem-solving capabilities.