Problem 48 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 5 m^{2}=15 m $$

Step-by-Step Solution

Answer

The solutions are \(m = 0\) and \(m = 3\).

Step 1: Move All Terms to One Side

First, move all terms to one side of the equation to set it to zero. The given equation is \(5m^2 = 15m\). Subtract \(15m\) from both sides to obtain:\[ 5m^2 - 15m = 0 \]

Step 2: Factor the Equation

Now, factor the equation \(5m^2 - 15m = 0\). Identify the greatest common factor (GCF) of the terms, which is \(5m\). Factor out \(5m\):\[ 5m(m - 3) = 0 \]

Step 3: Apply Zero Product Property

According to the zero product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). Apply this property to the factors:1. \(5m = 0\)2. \(m - 3 = 0\)

Step 4: Solve Each Factor Equation

Solve each equation obtained from the zero product property:1. For \(5m = 0\), divide both sides by 5: \[ m = 0 \]2. For \(m - 3 = 0\), add 3 to both sides: \[ m = 3 \]

Step 5: Write the Solution Set

The solutions for the equation \(5m^2 = 15m\) are the roots obtained from the previous step. Thus, the solution set is:\[ m = 0 \text{ or } m = 3 \]

Key Concepts

Factoring

Factoring is a powerful technique often used to solve quadratic equations. When you have a quadratic expression, you try to rewrite it in a simpler form by taking out common factors. This process not only simplifies the equation but also makes it easier to find solutions. In the equation given, which is initially written as \(5m^2 - 15m = 0\), the goal is to find the greatest common factor (GCF) of the terms. For \(5m^2 - 15m\), the GCF is \(5m\):

• \(5m^2\) and \(15m\) both have \(5m\) in common.
• When we factor \(5m\) out, the equation becomes \(5m(m - 3) = 0\).

Factoring transforms the quadratic equation into a product of terms, setting the stage for applying further methods to find the solution, such as the zero product property.

Zero Product Property

The zero product property is a fundamental concept in algebra that tells us about the nature of zero in multiplication. When you have the product of two numbers equaling zero, at least one of the numbers must also be zero.In mathematical terms, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). This property is incredibly useful when solving factored equations. For the equation \(5m(m - 3) = 0\), you can use the zero product property by setting each factor equal to zero separately:

• \(5m = 0\) implies \(m = 0\).
• \(m - 3 = 0\) implies \(m = 3\).

By breaking it down this way, solving the equation becomes straightforward as you isolate each variable and find its possible values.

Solution Set

The solution set of an equation contains all possible values that make the equation true. When solving quadratic equations, the solution set is made up of the values obtained from solving each factor individually. From the equation \(5m(m - 3) = 0\), once you apply the zero product property and solve for each factor, you find:

• For \(5m = 0\), the solution is \(m = 0\).
• For \(m - 3 = 0\), the solution is \(m = 3\).

Thus, the complete solution set for the given equation \(5m^2 = 15m\) is \(m = 0\) or \(m = 3\). This means both values satisfy the original equation and are verified solutions, meaning they are all the possible answers to this equation.