Problem 10 - College Algebra Essentials

Question

Solve each equation in Exercises \(1-14\) by factoring. $$ 5 x^{2}-20 x=0 $$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x = 0\) and \(x = 4\).

Step 1: Set the equation to equal zero

The equation is already in the form \(ax^2 + bx + c = 0\). The equation given is \(5x^2 - 20x = 0\)

Step 2: Factor out the greatest common factor

In our equation, the greatest common factor of \(5x^2\) and \(-20x\) is \(5x\). When we factor out \(5x\), we get \(5x (x - 4) = 0\)

Step 3: Apply the zero-product property

The zero-product property states that if \(ab = 0\), then \(a = 0\) or \(b = 0\). Applying this to our equation \(5x (x - 4) = 0\), we get two equations: \(5x = 0\) and \(x - 4 = 0\)

Step 4: Solve each equation

Solving the equations \(5x = 0\) and \(x - 4 = 0\) we get two solutions for \(x\). Solving \(5x = 0\) we divide both sides by \(5\), giving \(x = 0\). Solving \(x - 4 = 0\) we add \(4\) to both sides, giving \(x = 4\).

Key Concepts

Zero-Product Property

The Zero-Product Property is a fundamental concept in algebra. It means that if the product of two numbers is zero, then at least one of the numbers must be zero. This is written as: - If \(ab = 0\), then \(a = 0\) or \(b = 0\). This property is particularly useful when dealing with factored equations. Once you've factored the quadratic equation into a product of simpler expressions, you can use this property to find the solutions. For example, in the equation \(5x(x - 4) = 0\), the Zero-Product Property tells us that either \(5x = 0\) or \(x - 4 = 0\). Applying this gives you the potential solutions for \(x\). This property helps simplify finding roots of equations and is a powerful tool for solving quadratics.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor shared by two or more terms. When factoring quadratic equations, finding the GCF is often the first step. To find the GCF in a polynomial, examine the coefficients and variables of each term. In the equation \(5x^2 - 20x = 0\), both terms \(5x^2\) and \(-20x\) share a common factor of \(5x\). By factoring out \(5x\), you simplify the expression to \(5x(x - 4) = 0\). This reduction is crucial as it sets up the use of the Zero-Product Property by getting the equation ready for solving. Factoring out the GCF makes handling complex polynomials easier and more intuitive.

Quadratic Equation Solutions

Quadratic equations can usually be solved by factoring, using the quadratic formula, or completing the square. Each method has its proper time and place in problem-solving. If the quadratic is easy to factor, this method is usually the fastest and most straightforward. In this example, the equation \(5x^2 - 20x = 0\) is solved by factoring. After factoring the expression to \(5x(x - 4) = 0\) and applying the Zero-Product Property, two simple equations are solved: - First, \(5x = 0\) simplifies to \(x = 0\).- Second, solving \(x - 4 = 0\) gives \(x = 4\). These solutions mean that the roots of the quadratic equation are \(x = 0\) and \(x = 4\). Understanding these methods broadens your ability to tackle any quadratic equation efficiently, no matter its form.