Problem 53 - Elementary Algebra

Question

For the following problems, use the zero-factor property to solve the equations. $$ (2 m-1)(3 m+8)=0 $$

Step-by-Step Solution

Answer

Answer: The possible solutions for the equation are \(m = \frac{1}{2}\) and \(m = -\frac{8}{3}\).

Step 1: Write down the given equation

We are given the equation: $$ (2m-1)(3m+8) = 0 $$

Step 2: Apply the zero-factor property

Using the zero-factor property, we can set each factor equal to zero: $$ 2m - 1 = 0 \quad \text{or} \quad 3m + 8 = 0 $$

Step 3: Solve the first equation

To solve \(2m - 1 = 0\), we add 1 to both sides and then divide by 2: $$ 2m = 1 \\ m = \frac{1}{2} $$

Step 4: Solve the second equation

To solve \(3m + 8 = 0\), we subtract 8 from both sides and then divide by 3: $$ 3m = -8 \\ m = -\frac{8}{3} $$

Step 5: State the solutions

The possible solutions for the equation are: $$ m = \frac{1}{2} \quad \text{or} \quad m = -\frac{8}{3} $$

Key Concepts

Solving Equations

When we talk about solving equations, we're referring to the process of finding the values that satisfy a given mathematical statement. In this context, the equation given is \((2m-1)(3m+8) = 0\). The objective is to determine the values of \(m\) that make this equation true. Often, in algebra, equations can have multiple solutions.

The equation above is a product of two linear expressions. Using the zero-factor property, we understand that if the product of two or more terms is zero, at least one of the terms must be zero. Therefore, solving the equation involves setting each separate term equal to zero. This is a fundamental principle used to break down complex problems into simpler parts, allowing us to tackle each factor individually for potential solutions.

Factorization

Factorization is a process in mathematics used to break down complex expressions into simpler, multiple terms whose product is the original expression. In the given exercise, the expression \((2m-1)(3m+8)\) has already been factorized. This means it has been written as a product of two simpler expressions.

Factorization can help in identifying zeros of the equation, which provides solutions based on the zero-product property. Recognizing factorized equations allows us to apply strategies such as setting each term separately to zero. This method efficiently streamlines the problem-solving process, allowing students to handle polynomial equations by tackling individual linear factors.

Algebraic Solutions

Finding algebraic solutions involves manipulation of equations to find variable values. After applying the zero-factor property to the factors \(2m-1\) and \(3m+8\), we solve each with algebraic operations.

• For \(2m - 1 = 0\), we add 1 to isolate the term with the variable, resulting in \(2m = 1\), then divide each side by 2 to get \(m = \frac{1}{2}\).
• For \(3m + 8 = 0\), we subtract 8 to isolate terms with the variable, resulting in \(3m = -8\), then divide each side by 3 to solve for \(m\), yielding \(m = -\frac{8}{3}\).

This method shows how breaking down algebraic expressions into simpler components easily guides us to the solutions, making complex equations approachable and solvable.