Question

Write a mathematical model for the number problem, and solve the problem. Find two consecutive integers whose product is five less than the square of the smaller number.

Short answer

The two consecutive integers whose product is five less than the square of the smaller number are -5 and -4.

Step-by-step solution

Define Terms

Define the smaller integer as \( x \). Thus, the next consecutive integer will be \( x + 1 \).

Formulate the Mathematical Model

Re-code the problem's conditions into a mathematical equation. According to the problem, the product of the two consecutive integers is 5 less than the square of the smaller integer. This can be written as \( x(x + 1) = x^2 - 5 \).

Simplify the Equation

Simplify the above equation and collect all the terms on one side to facilitate further calculations. This simplifies to \( x^2 + x = x^2 - 5 \), which further simplifies to \( x = -5 \) after cancelling out the \( x^2 \) on both sides.

Find the Consecutive Integer

As \( x = -5 \), so the next consecutive integer will be \( x + 1 = -4 \).

Verification

Verify if the integers \( -5 \) and \( -4 \) satisfy the original condition set in the problem. \((-5)(-4) = -5^2 - 5 \), or \(20 = 25 - 5 = 20\), which is a true statement. Thus, the integers \( -5 \) and \( -4 \) are indeed the solution to the problem.

Key concepts

Mathematical Modeling

Mathematical modeling involves translating real-world scenarios into the language of mathematics to facilitate problem-solving. In the context of consecutive integers, we create algebraic expressions that represent the problem's conditions.

For the given exercise, we define our unknowns in algebraic terms: let the smaller integer be represented by the variable \( x \). We can express the next consecutive integer as \( x + 1 \), since consecutive integers have a difference of one.

The problem states that the product of these integers is five less than the square of the smaller one. This relationship is encapsulated in the equation \( x(x + 1) = x^2 - 5 \). It's essential to translate the problem accurately because a proper model is the foundation for finding a solution. The model not only encapsulates the given situation but also guides us through the equation solving process.

Equation Solving

Equation solving is the process of finding the value(s) of the variable that make an equation true. It involves manipulating the equation while preserving its equality until the variable is isolated.

Using our model, \( x(x + 1) = x^2 - 5 \), we start by simplifying to get all terms involving \( x \) on one side, leading to \( x^2 + x = x^2 - 5 \). By subtracting \( x^2 \) from each side, we isolate \( x \) to obtain \( x = -5 \).

This step, crucial in solving our problem, involves applying algebraic rules consistently to simplify the equation and isolate the variable which is the key to unlocking the problem. With the variable determined, we can move on to find the consecutive integer and verify our solution.

Algebraic Equations

Algebraic equations are the cornerstone of solving mathematical problems involving unknown values. They are statements of equality with variables that stand in for unknown quantities.

In our problem, the algebraic equation \( x(x + 1) = x^2 - 5 \) sets up a relationship between the products of consecutive integers and a given square function. The true power of algebraic equations is revealed when we manipulate these equations to find the values of the unknowns.

Understanding how to construct and solve algebraic equations is pivotal when approaching a variety of math problems. Success in solving these problems often hinges on recognizing patterns, abstracting the relationships involved, and applying algebraic principles to manipulate and simplify expressions.