Problem 4 - 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. $$x^{2}=15 x$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = 15\).

Step 1: Write Equation in Standard Form

First, we need to rewrite the given quadratic equation in standard form. The standard form is \(ax^2 + bx + c = 0\). Start with the equation \(x^2 = 15x\). By subtracting \(15x\) from both sides, we obtain \(x^2 - 15x = 0\).

Step 2: Factor the Quadratic Expression

In this step, we factor the quadratic expression \(x^2 - 15x\). Look for the greatest common factor (GCF) between the terms. The GCF here is \(x\), so factor it out: \(x(x - 15) = 0\).

Step 3: Apply Zero Product Property

Use the zero product property, which states that if \(a \cdot b = 0\), then \(a = 0\) or \(b = 0\). Apply it to the factored form: \(x = 0\) or \(x - 15 = 0\).

Step 4: Solve Each Equation

Solve for \(x\) in both equations obtained in Step 3. For \(x = 0\), the solution is simply \(x = 0\). For \(x - 15 = 0\), add \(15\) to both sides to get \(x = 15\). Therefore, the solutions are \(x = 0\) and \(x = 15\).

Key Concepts

Factoring

Factoring is a method used to simplify expressions or solve equations by rewriting them as a product of their factors. In the context of quadratic equations, factoring means expressing the equation in the form of a product of binomials or a simpler expression.

When solving a quadratic equation by factoring, the first step often involves finding the greatest common factor (GCF) from the terms of the equation. For example, consider the quadratic expression \(x^2 - 15x\). Here, the GCF is \(x\), which can be factored out to get \(x(x - 15) = 0\).

It's crucial to be skillful in identifying common factors and using them to simplify the equation.

• Start by examining each term in the equation.
• Determine what variables or numbers can be factored out.
• Rewrite the expression as a product of factors.

Factoring simplifies the equation and sets up the groundwork for applying the zero product property.

Zero Product Property

The zero product property is a fundamental concept often used after factoring a quadratic equation. This property states that if a product of two numbers (or expressions) is zero, at least one of the factors must be zero. So, if \(a \cdot b = 0\), then either \(a = 0\) or \(b = 0\).

After factoring the equation \(x(x - 15) = 0\), you apply the zero product property by setting each factor equal to zero. This results in two simple equations: \(x = 0\) and \(x - 15 = 0\). Solving these equations individually gives the solutions to the original quadratic equation.

This property is particularly powerful because it breaks down potentially complex quadratic equations into much simpler linear equations, making it easier to find the roots. Remember:

• If a product equals zero, check each factor individually.
• Set each factor equal to zero and solve.

This step is essential for finding the solutions efficiently after successful factoring.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is essential for systematic solving and is typically expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. This form helps in organizing the equation to apply various methods such as factoring, completing the square, or using the quadratic formula.

In the given problem, the original equation \(x^2 = 15x\) needs to be rearranged into standard form. By subtracting \(15x\) from both sides, it becomes \(x^2 - 15x = 0\). This transformation is critical because only then can we properly apply techniques like factoring to solve the equation.

Here are some basic steps to write an equation in standard form:

• Ensure all terms are on one side of the equation, with zero on the other.
• Align and simplify terms in descending powers of \(x\).
• Check the equation structure against the standard form.

Organizing the equation into standard form makes subsequent steps more straightforward and brings clarity to the solving process.