Solving and Factoring Quadratic Equations: A Comprehensive Guide

Understanding how to solve and factor quadratic equations is a fundamental skill in algebra. This article will explore how to properly factor the quadratic equation 2x^2 - 8x - 6 and provide a method to solve the equation 2x^2 - 8x - 6 0. We'll discuss various factorization methods and show step-by-step solutions.

Factorization Methods in Quadratic Equations

Quadratic equations are expressions of the form ax^2 bx c 0. To solve such equations, one common method is factoring. For the given equation 2x^2 - 8x - 6, we need to find factors that satisfy the equation's structure.

Factorizing by Grouping and Pulling Out the Greatest Common Factor (GCF)

Let's start by factoring out the greatest common factor from the equation. Notice that all the terms have a common factor of 2:

2x^2 - 8x - 6 2(x^2 - 4x - 3)

Next, we focus on factoring the quadratic expression x^2 - 4x - 3. To do this, we need to find two numbers that multiply to give -3 and add up to -4. The factors of -3 that add up to -4 are -1 and -3. Therefore, we can write:

x^2 - 4x - 3 (x - 1)(x - 3)

Thus, the original equation can be factored as:

2x^2 - 8x - 6 2(x - 1)(x - 3)

Solving the Equation 2x^2 - 8x - 6 0

To find the solutions of the equation 2x^2 - 8x - 6 0, we set the factored form equal to zero:

2(x - 1)(x - 3) 0

Since the product of the factors is zero, at least one of the factors must be zero. Therefore:

(x - 1) 0 or (x - 3) 0

Solving each equation separately:

x - 1 0 implies x 1

x - 3 0 implies x 3

Thus, the solutions to the equation 2x^2 - 8x - 6 0 are x 1 and x 3.

Using Technology to Solve Equations

If you're struggling with complex equations, tools like the CameraMath app can be incredibly useful. By clicking the “Show Steps” tab, you can get a step-by-step explanation of the process. Join the platform to access more resources and support for your math needs.

Practice and Learning

To truly master quadratic equations, it's important to practice regularly. Understanding the methods and steps involved is crucial. If you're posting multiple questions without engaging with the answers, you may not learn as effectively. Use resources and explanations provided in previous answers to enhance your learning process.

Stay curious and keep practicing, and you will see significant improvement in your skills.