Problem 55 - Algebra for College Students

Question

Set up an equation and solve each problem. Find two consecutive integers whose product is 72 .

Step-by-Step Solution

Answer

The consecutive integers are 8 and 9 or -9 and -8.

Step 1: Define the Variables

Let's define the first integer as \( x \). Since the integers are consecutive, the next integer will be \( x + 1 \).

Step 2: Set Up the Equation

The problem states that the product of these two integers is 72. Therefore, we can set up the equation: \( x(x + 1) = 72 \).

Step 3: Simplify the Equation

Expand the equation: \( x^2 + x = 72 \).

Step 4: Rearrange to Form a Quadratic Equation

Subtract 72 from both sides to get a standard form quadratic equation: \( x^2 + x - 72 = 0 \).

Step 5: Factor the Quadratic Equation

Factor the quadratic equation to find the integers: \((x + 9)(x - 8) = 0\).

Step 6: Solve for x

Set each factor equal to zero and solve for \( x \):- \( x + 9 = 0 \) gives \( x = -9 \),- \( x - 8 = 0 \) gives \( x = 8 \).

Step 7: Determine Consecutive Integers

For \( x = 8 \), the integers are 8 and 9.For \( x = -9 \), the integers are -9 and -8.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a type of polynomial equation that involve terms up to the second degree. This means they have a variable raised to the power of two, typically represented as:\[x^2 + bx + c = 0\]The "\(x^2\)" denotes the second-degree term. Quadratic equations are essential because they help us find two unknown values, like the consecutive integers in this problem.

• The standard form of a quadratic equation is \(ax^2 + bx + c = 0\).
• Here, "a," "b," and "c" are constants. In many quadratic problems, one of these variables will be zero, simplifying the equation.

To solve the exercise, we rearrange the product equation into this standard form by subtracting 72 from both sides. This ensures the equation is ready for factoring and finding solutions.

Mastering Factoring

Factoring is a helpful method for solving quadratic equations when they can be broken down into simpler expressions multiplied together. The factored form of our original quadratic equation, \(x^2 + x - 72 = 0\), becomes \((x + 9)(x - 8) = 0\).The process of factoring can be likened to reverse multiplication, where:

• We identify two numbers that multiply to the constant term (\(-72\)) and add to the middle term coefficient (1).
• In this exercise, 9 and -8 fit perfectly, because \(9 \cdot -8 = -72\) and \(9 + (-8) = 1\).

By expressing the quadratic equation this way, we can easily set each factor to zero to find the solutions for \(x\). This reveals the possible values of the first consecutive integer.

Unraveling Integer Solutions

When solving quadratic equations, particularly ones like this where you want consecutive integers, you’re often interested in integer solutions. These are solutions or values of \(x\) in the factored equation that are whole numbers.

• Integer solutions are crucial because they match the context of real-life problems involving countable items, such as finding consecutive integers.
• In "\((x + 9) = 0\) and "\((x - 8) = 0\), solving for \(x\) reveals that \(x = -9\) and \(x = 8\). Both are integers.

These solutions indicate the first possible integer of our consecutive pair. Once known, simply add one to these values to find the second integer. Thus, the consecutive integers can be \(8, 9\) or \(-9, -8\).

Product of Integers and Its Role

In this problem, knowing that two consecutive integers multiply to a specific product lays the groundwork for setting up an equation. The product of integers is derived from multiplying two numbers.

• Products that refer to consecutive integers come from expressions like \(x(x + 1) = 72\), where we're finding two numbers close together that reach this specific value.
• The importance lies in identifying numbers that both solve the equation and fulfill the consecutive condition.

When these integers multiply to a given number, like 72 in this problem, they efficiently lead us to the integer solutions needed. This "multiplicative relationship" is often used in problems involving ages, areas, and other settings where specific numerical relationships are needed.