Problem 85 - Intermediate Algebra

Question

Solve each equation. See Sections 2.1 and 5.8. $$ (x-6)(2 x+1)=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 6\) and \(x = -\frac{1}{2}\).

Step 1: Understand the Zero Product Property

The Zero Product Property states that if a product of factors equals zero, then at least one of the factors must be zero. In this problem, we have two factors: \((x-6)\) and \((2x+1)\). If their product is zero, then either \((x-6)=0\) or \((2x+1)=0\).

Step 2: Solve Each Factor for Zero

Set each factor equal to zero and solve for \(x\).1. Solve \(x-6=0\): - Add 6 to both sides: \(x = 6\).2. Solve \(2x+1=0\): - Subtract 1 from both sides: \(2x = -1\). - Divide both sides by 2: \(x = -\frac{1}{2}\).

Step 3: Verify the Solutions

Verify the solutions by substituting them back into the original equation. 1. Substitute \(x=6\): - LHS: \((6-6)(2 \times 6 + 1) = 0 \times 13 = 0\).2. Substitute \(x=-\frac{1}{2}\): - LHS: \((-\frac{1}{2}-6)(2 \times -\frac{1}{2} + 1) = -\frac{13}{2} \times 0 = 0\).Both substitutions validate the solutions.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are fundamental in algebra and arise in various mathematical contexts. A typical quadratic equation has the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In this form, \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term. Quadratic equations can describe parabolic graphs that open upwards or downwards depending on the sign of \(a\).

In the exercise provided, although the equation appears in factored form, it's essential to link it back to the quadratic concept. If expanded, the expression \((x-6)(2x+1)\) is indeed a quadratic equation as it involves the product of two linear terms. This highlights the powerful technique of factoring in solving quadratic equations.

Applications of quadratics span various fields such as physics for describing projectiles, economics for modeling profit and cost, and many other areas. Understanding the basic structure and methods to solve quadratic equations is a crucial skill in math.

The Art of Factoring

Factoring is a technique used to rewrite an expression as the product of its factors. For quadratic expressions, factoring involves representing \(ax^2 + bx + c = 0\) in the form \((px + q)(rx + s) = 0\). This form makes it easy to identify the roots of the quadratic equation.

In our example, the expression \((x-6)(2x+1) = 0\) is already factored. The goal is to find two numbers whose product equals zero, allowing us to apply the Zero Product Property effectively. Factoring serves as a bridge to simplifying and solving complex equations, making it a fundamental skill to master.

Reasons to use factoring include:

• Simplification: Breaks down complex polynomials into manageable parts.
• Solving: Provides a direct method to uncover solutions or roots.
• Insight: Offers deeper understanding of the equation's structure and components.

Mastering factoring equips students to tackle a broad range of mathematical challenges.

Solving Equations Like an Expert

Solving equations is a cornerstone of algebra that involves finding the value that satisfies the given equation. When solving equations, especially quadratic ones via factoring, we rely on the Zero Product Property. This property states that if the product of two factors is zero, then at least one of the factors must be zero.

Let's review the process applied in the solution:

• Identify the factors: For \((x-6)(2x+1)=0\), we have the factors \((x-6)\) and \((2x+1)\).
• Apply the Zero Product Property: Set each factor equal to zero. This gives two equations: \(x-6=0\) and \(2x+1=0\).
• Solve each equation:
  • For \(x-6=0\), adding 6 to both sides yields \(x=6\).
  • For \(2x+1=0\), subtracting 1 and dividing by 2 gives \(x=-\frac{1}{2}\).

To ensure accuracy, substitute these values back into the original equation to verify that they yield zero. This verification process confirms that no errors were made during calculations.

By solving equations systematically, you can confidently find solutions and deepen your understanding of mathematical relationships.