Problem 11 - Algebra for College Students

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. $$x^{2}-19 x+84=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 7\) and \(x = 12\).

Step 1: Identify the Quadratic Equation

The given quadratic equation is \(x^2 - 19x + 84 = 0\) and is in the standard form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -19\), and \(c = 84\).

Step 2: Determine Factors of the Constant Term

To solve by factoring, look for two numbers that multiply to 84 (the constant term \(c\)) and add to -19 (the linear coefficient \(b\)).

Step 3: Find Suitable Factor Pairs

The factor pairs of 84 that add up to -19 are -7 and -12. Thus, \(-7\times -12 = 84\) and \(-7 + (-12) = -19\).

Step 4: Write the Factored Form

Rewrite the quadratic equation by factoring it: \(x^2 - 19x + 84 = (x - 7)(x - 12) = 0\).

Step 5: Apply the Zero Product Property

According to the zero product property, if \((x - 7)(x - 12) = 0\), then \(x - 7 = 0\) or \(x - 12 = 0\).

Step 6: Solve Each Equation for x

Solving \(x - 7 = 0\) gives \(x = 7\). Solving \(x - 12 = 0\) gives \(x = 12\). Thus, the solutions to the equation are \(x = 7\) and \(x = 12\).

Key Concepts

Factoring Techniques

Factoring is a powerful technique used to solve quadratic equations. It involves rewriting the quadratic expression as a product of two binomials. For the given equation, \(x^2 - 19x + 84 = 0\), we aim to find two binomials, \((x - p)(x - q)\). These binomials multiply to give the original quadratic expression.

The process essentially involves:

• Identifying two numbers, \(p\) and \(q\), such that their product equals the constant term, \(c = 84\), and their sum equals the linear coefficient, \(b = -19\).
• In this specific example, we find that \(p = 7\) and \(q = 12\) are the numbers that fit these conditions since they multiply to 84 and add to 19 (considering the negative signs).

By rearranging, we write the quadratic equation in its factored form: \((x - 7)(x - 12) = 0\). This factored expression is essential as it sets up the next step where we use the zero product property.

Zero Product Property

The zero product property is a fundamental principle in algebra that applies when a product of factors equals zero. It states that if \(ab = 0\), then \(a = 0\) or \(b = 0\).

In the quadratic equation \((x - 7)(x - 12) = 0\), this property is leveraged:

• If either \((x - 7)\) or \((x - 12)\) is zero, the original product will also be zero. Therefore, we consider each factor individually:
• Set the first factor to zero: \(x - 7 = 0\), which simplifies to \(x = 7\).
• Set the second factor to zero: \(x - 12 = 0\), which simplifies to \(x = 12\).

Using the zero product property simplifies breaking down a complex quadratic equation into manageable pieces, allowing us to solve for the variable quickly.

Solving Quadratic Equations

Solving quadratic equations like \(x^2 - 19x + 84 = 0\) involves several steps that systematically break down the problem:

• First, write the equation in standard form \(ax^2 + bx + c = 0\).
• Second, apply factoring techniques to express the quadratic as the product of two binomials.
• Next, apply the zero product property to split the equation into simpler linear equations.
• Solve these linear equations for the variable \(x\).

The solutions we found are \(x = 7\) and \(x = 12\). These values satisfy the original quadratic equation as plugging them back in results in zero. Mastering these step-by-step techniques in solving quadratic equations is key to understanding more complex algebraic concepts later on. By practicing these methods, students can enhance their proficiency in solving a wide range of mathematical problems.