Solving Quadratic Equations by Factorization: A Detailed Guide

Quadratic equations are a fundamental part of algebra and are widely used in various fields such as physics, engineering, and economics. One of the common methods to solve these equations is by factorization. In this guide, we will explore how to solve a specific quadratic equation using factorization, and provide a step-by-step process for doing so.

Understanding Quadratic Equations

A quadratic equation is an equation that can be written in the standard form:

ax2 bx c 0

The variable x is of degree two, meaning it is squared. In this article, we will focus on solving quadratic equations that can be factored.

Example: Solving 5x2 3

Let's consider the equation:

5x2 3

First, rewrite it as a quadratic equation equal to zero:

5x2 - 3 0

Now, let's factor this equation by following a step-by-step process.

Step-by-Step Process

Step 1: Re-writing the Equation

Our goal is to have the equation in the form ax2 - c 0:

5x2 - 3 0

Step 2: Factoring the Greatest Common Factor

The greatest common factor (GCF) in this case is 1, as both terms are already in their simplest form. However, if we were to have a common factor, we would factor it out. In this example, the equation is already ready for the next step, which is factoring the equation into two binomials.

Step 3: Factoring the Equation

The next step is to factor the quadratic equation into two binomials. In this case:

5x2 - 3 5x - 3 * (x - 2) 0

However, 5x - 3 is not a perfect factor, so we need to look for roots that fit the pattern of the quadratic equation. In this specific example, the equation can be factored as:

(5x - 6)(x - 1) 0

Step 4: Applying the Zero-Product Property

According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero:

5x - 6 0 or x - 1 0

Solve each factor for x:

Therefore, the solutions to the equation 5x2 - 3 0 are x 6/5 and x 1.

Common Misconceptions and FAQs

Q: What if the 2 is an exponent?

Confusion often arises when the number 2 appears as an exponent, leading to the equation being written in a different form. In this case, the equation should be treated as a quadratic equation and solved using the appropriate methods. For example:

5x2 3 can be rewritten as:

5x2 - 3 0

Then proceed with the steps outlined in the guide.

Q: How do we factor quadratic expressions?

Factoring quadratic expressions involves finding two binomials whose product equals the quadratic expression. This often requires finding two numbers that multiply to give the constant term (c) and add to give the coefficient of the middle term (b).

Q: What is the zero-product property?

The zero-product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property is used in solving quadratic equations by factoring.

Conclusion

Understanding the process of solving quadratic equations by factorization is crucial for anyone wanting to tackle problems in mathematics and related fields. By following these steps, you can easily solve any quadratic equation that can be factored. Whether you are a student, a professional, or simply someone interested in mathematics, this guide provides a clear and concise method for solving such equations.