Problem 29 - 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 \(b=0\). $$x^{2}=16 k^{2} x$$

Step-by-Step Solution

Answer

The solutions for \(x\) are \(x = 0\) and \(x = 16k^2\).

Step 1: Rewrite the Equation

The given equation is \(x^2 = 16k^2x\). First, we need to rewrite this equation in standard quadratic form \(ax^2 + bx + c = 0\). Subtract \(16k^2x\) from both sides to get \(x^2 - 16k^2x = 0\).

Step 2: Factor the Common Term

Notice that there is a common factor of \(x\) in both terms on the left-hand side of the equation. Factor \(x\) out: \(x(x - 16k^2) = 0\).

Step 3: Apply the Zero Product Property

According to the zero product property, if \(ab = 0\), then \(a = 0\) or \(b = 0\). Apply this property to the factored equation \(x(x - 16k^2) = 0\). Thus, either \(x = 0\) or \(x - 16k^2 = 0\).

Step 4: Solve for x

We have two simple equations: \(x = 0\) which is already solved, and \(x - 16k^2 = 0\). Solve \(x - 16k^2 = 0\) by adding \(16k^2\) to both sides to get \(x = 16k^2\).

Key Concepts

Zero Product Property

In algebra, a fundamental tool we often rely on is the zero product property. 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 \(a \times b = 0\), then either \(a = 0\) or \(b = 0\).
This concept is particularly important in solving quadratic equations by factoring. It helps in breaking down complex problems into simpler parts.
When we encounter a factored equation, we apply the zero product property to set each factor equal to zero, thus simplifying our path to finding the solutions. By solving each resulting equation, we find the values of the variables that satisfy the original equation.
Mastering the zero product property is essential for solving quadratic equations efficiently. It saves time and can be applied to a variety of math problems.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, meaning they have the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\).
Quadratic equations are crucial in various fields such as physics, engineering, and finance because they can model real-world scenarios like projectile motion or economic profit maximization.
There are several methods to solve quadratic equations, including:

• Factoring
• Completing the square
• Using the quadratic formula

Each method involves different steps and is chosen based on the form of the quadratic equation.
Understanding the structure of quadratic equations allows us to employ these methods efficiently and to choose the most appropriate strategy for a given equation.

Solving Equations by Factoring

Solving equations by factoring involves breaking down a complex equation into simpler, multiplicative components. This method is essential for quadratic equations that can be expressed as a product of linear factors.
To solve by factoring, follow these steps:

• Write the equation in standard form \(ax^2 + bx + c = 0\).
• Identify any common factors across terms and factor them out.
• Rewrite the quadratic as a product of two binomials.
• Apply the zero product property by setting each binomial to zero.
• Solve each resulting simple equation to find the values of \(x\).

Factoring helps us find solutions quickly and is especially straightforward when the quadratic equation is easily factored.
The beauty of this method lies in its simplicity, making it a favorite among students and mathematicians alike for solving basic quadratic equations efficiently.