Problem 12 - Intermediate Algebra

Question

Solve each of the quadratic equations by factoring and applying the property, \(a b=0\) if and only if \(a=0\) or \(b=0\). If necessary, return to Chapter 3 and review the factoring techniques presented there. $$ x^{2}-21 x+104=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 8\) and \(x = 13\).

Step 1: Identify and Rearrange the Equation

The given quadratic equation is in the standard form as follows: \[ x^2 - 21x + 104 = 0 \]which can be readily used for factoring.

Step 2: Identify a and c for Factoring

In factorable quadratics of the form \( ax^2 + bx + c = 0 \), identify the coefficients:- Coefficient of \(x^2\) (or \(a\)) is 1.- Coefficient of \(x\) (or \(b\)) is -21.- Constant term (or \(c\)) is 104.We are looking for two numbers that multiply to 104 and add to -21.

Step 3: Find Factor Pairs of 104

List potential pairs of integers that multiply to 104: (1, 104), (2, 52), (4, 26), and (8, 13). Consider signs as needed, given that our middle term is negative.

Step 4: Choose Correct Pair of Factors

Among the pairs (8, 13), check if any pair adds up to -21. The pair (-8, -13) adds up to -21 and multiplies to 104, satisfying both conditions.

Step 5: Rewrite in Factored Form

Rewrite the quadratic in its factored form using the numbers found:\[ (x - 8)(x - 13) = 0 \]

Step 6: Apply Zero Product Property

According to the zero product property, if \( (x - 8)(x - 13) = 0 \), then:\[ x - 8 = 0 \quad \text{or} \quad x - 13 = 0 \]

Step 7: Solve the Equations

Solve each equation separately:1. \(x - 8 = 0\) gives \(x = 8\).2. \(x - 13 = 0\) gives \(x = 13\).

Step 8: State the Solutions

The solutions to the quadratic equation \(x^2 - 21x + 104 = 0\) are \(x = 8\) and \(x = 13\).

Key Concepts

Factoring

Factoring is a crucial method for solving quadratic equations. The main idea is to express the quadratic equation as a product of two factors set to zero. Consider a standard quadratic equation in the form \( ax^2 + bx + c = 0 \).

• The goal is to find two binomials that multiply to form the original quadratic equation.
• In our example, \( x^2 - 21x + 104 = 0 \), we looked for two numbers that both add up to \(-21\) and multiply to give 104.

This process involved finding factor pairs of the constant term (104), such as (1, 104), (2, 52), (4, 26), and (8, 13). Among these, the pair (-8, -13) not only multiplies to 104 but also adds up to -21, perfectly aligning with the coefficients of the quadratic equation. Rewriting the equation in its factored form \((x - 8)(x - 13) = 0\) is the next step in solving.

Zero Product Property

The Zero Product Property is an essential rule in algebra that states if the product of two factors is zero, then at least one of the factors must be zero. Once a quadratic equation is factored, this property is used to find the solutions.

• In the factored equation \((x - 8)(x - 13) = 0\), the Zero Product Property tells us \(x - 8 = 0\) or \(x - 13 = 0\).
• This means the solutions can be found by setting each factor equal to zero separately.

This property simplifies solving equations significantly, as it turns the task of solving a quadratic equation into solving two simpler equations. It’s a straightforward yet powerful tool.

Quadratic Solutions

Solving a quadratic equation means finding the values of \(x\) that make the equation true. With the equation \(x^2 - 21x + 104 = 0\) rewritten in the form of \((x - 8)(x - 13) = 0\), the next step is to find the values of \(x\).

• From \(x - 8 = 0\), solving gives \(x = 8\).
• From \(x - 13 = 0\), solving gives \(x = 13\).

These solutions, \(x = 8\) and \(x = 13\), mean if you substitute either back into the original quadratic equation, they will satisfy the equation. The outcome illustrates how factoring and applying the Zero Product Property effectively unveil the square roots or solutions of the quadratic equation.