Problem 27 - Algebra for College Students

Question

Solve each equation for \(x\) by factoring and applying the property, \(a b=0\) if and only if \(a=0\) or \(b=0\). $$x^{2}-5 k x=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = 5k\).

Step 1: Identify the common factor

The given equation is \(x^2 - 5kx = 0\). Notice that both terms include \(x\). The common factor that can be factored out is \(x\).

Step 2: Factor out the common factor

Factor \(x\) from each term in the equation: \(x(x - 5k) = 0\).

Step 3: Use the Zero Product Property

According to the Zero Product Property, if a product of two factors equals zero (\(ab = 0\)), then at least one of the factors must be zero. Apply this property to \(x(x - 5k) = 0\). This gives two possible equations: \(x = 0\) or \(x - 5k = 0\).

Step 4: Solve each equation

Solve each of the equations from Step 3: 1. \(x = 0\) is already solved.2. Solve \(x - 5k = 0\) for \(x\): Add \(5k\) to both sides to obtain \(x = 5k\).

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental concept in algebra, especially when dealing with quadratic equations. This property states that if the product of two numbers (or expressions) is zero, then at least one of the numbers must be zero. In mathematical terms, if \(ab = 0\), then \(a = 0\) or \(b = 0\). This principle is crucial because it allows us to break down complex equations into simpler parts that are easier to solve.
Applying the Zero Product Property is often the final step when solving factored equations. Once an expression is factored, each factor could potentially be zero, leading to straightforward linear equations. The simplicity of this property makes it a powerful tool for finding the solutions of quadratic equations, as seen in the original exercise.

Common Factor

Finding a common factor is the first step when factoring an equation. It involves identifying terms within the equation that share a common multiple. In the given equation \(x^2 - 5kx = 0\), both terms, \(x^2\) and \(-5kx\), include the variable \(x\). Therefore, \(x\) is considered the common factor.
Once a common factor is found, you can factor it out of the equation. This simplifies the equation by rewriting it as a product,

• \(x(x - 5k) = 0\)

Factoring is an essential procedure that sets up the equation for the Zero Product Property. Without factoring out the common factor, it would be impossible to break the equation into simpler components that can be solved using the Zero Product Property.

Solving Quadratic Equations

Solving quadratic equations often requires transforming the equation into a form where it can easily be factored or using the Zero Product Property. In the given example, the quadratic equation \(x^2 - 5kx = 0\) is already conducive to factoring due to the presence of \(x\) as a common factor.
The steps to solve this include:

• Identifying and factoring out the common factor.
• Applying the Zero Product Property to the factored expression.
• Solving the resulting simpler linear equations.

In this case:

• Factoring gives us \(x(x - 5k) = 0\).
• Using the Zero Product Property: \(x = 0\) or \(x - 5k = 0\).
• Solve: \(x = 0\) and \(x = 5k\).

By following these steps, we solve the quadratic equation and find all possible values of \(x\). This method is not only effective but also lays the groundwork for solving more complex equations in algebra.