Problem 10 - 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. Find two consecutive odd integers whose product is 143

Step-by-Step Solution

Answer

The pairs are (11, 13) and (-13, -11).

Step 1: Represent the odd integers

Let the first odd integer be represented by \( x \). Since the next consecutive odd integer would be 2 units more than the first, represent the second odd integer as \( x + 2 \).

Step 2: Set up the equation

According to the problem, the product of these two consecutive odd integers is 143. Set up the equation as follows: \( x(x + 2) = 143 \).

Step 3: Solve the quadratic equation

Expand and rearrange the equation: \( x^2 + 2x - 143 = 0 \). Solve this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = -143 \).

Step 4: Find the roots

Calculate the discriminant: \( b^2 - 4ac = 2^2 - 4(1)(-143) = 4 + 572 = 576 \). Now find the roots: \( x = \frac{-2 \pm \sqrt{576}}{2} \). This simplifies to \( x = \frac{-2 \pm 24}{2} \).

Step 5: Determine the integer values

Solve for the values of \( x \): \( x = \frac{22}{2} = 11 \) and \( x = \frac{-26}{2} = -13 \). So, the two possible pairs of consecutive odd integers are \( 11 \) and \( 13 \), or \( -13 \) and \( -11 \).

Step 6: Verify the products

Verify the products: \( 11 \times 13 = 143 \) and \( -13 \times -11 = 143 \). Both pairs satisfy the condition given in the problem.

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). They play an essential role in algebra and can describe many real-world scenarios.

To solve a quadratic equation, we can use the **quadratic formula**: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula is derived from completing the square method.

In our exercise, we derived the quadratic equation \( x^2 + 2x - 143 = 0 \) after setting up the product of two consecutive odd integers.

Let's break down how this relates to the quadratic formula:

• - **a** is the coefficient of \( x^2 \), which is 1 in this case.
• - **b** is the coefficient of \( x \), which is 2.
• - **c** is the constant term, which is -143 here.

By using \( a = 1 \), \( b = 2 \), and \( c = -143 \), we calculated the discriminant \( b^2 - 4ac \) and then applied the values to the formula to find the roots.

Understanding the quadratic formula is crucial for solving many algebraic problems!

Integer Properties

Integers are whole numbers that can be positive, negative, or zero. Some important properties of integers are:

• - **Consecutive Integers**: Numbers that follow each other in order without any gaps. For example, 3 and 4 are consecutive integers.
• - **Consecutive Odd/Even Integers**: Successive integers that are either all odd or all even. For example, 1 and 3 are consecutive odd integers. In the exercise, we use consecutive odd integers \( x \) and \( x + 2 \).
• - **Addition and Multiplication**: Addition or multiplication of two integers always results in an integer. For instance, in the product \( x(x + 2) = 143 \), both \( x \) and \( x + 2 \) must be integers for the product to be 143, which is also an integer.

These properties make integers predictable and manageable, which is why they are extensively used in algebra and other areas of mathematics.

By leveraging these properties, we simplified solving our original problem involving the product of consecutive odd integers.

Factoring

Factoring is the process of breaking down an expression into simpler 'factors' that, when multiplied together, yield the original expression. It's a critical skill in solving quadratic equations.

In the given exercise, we factorized the product equation: \( x(x + 2) = 143 \) to \( x^2 + 2x - 143 = 0 \).

Since this equation was a standard quadratic form, solving it involved:

• - **Expanding the Equation**: Multiplying terms to express it in the form \( ax^2 + bx + c = 0 \).
• - **Using the Quadratic Formula**: Once expanded, applying the quadratic formula to find the roots.
• - **Verification**: After calculating the potential values for \( x \), substituting back to verify if the product indeed equals 143.

Factoring simplifies equations and reveals their roots or possible solutions. It's a fundamental algebraic tool for solving complex mathematical problems.

With practice, recognizing patterns and applying factoring techniques becomes intuitive!