How to Solve a Quadratic Equation Using Factorization: A Comprehensive Guide

Quadratic equations are a fundamental part of algebra and are encountered in various fields such as physics, engineering, and mathematics. A quadratic equation is generally expressed in the form ax^2 bx c 0. In this article, we will explore how to solve the given quadratic equation x^2 - 7x 2x^2 - 8x - x - 4 using the factorization method.

Steps to Solve a Quadratic Equation Using Factorization

To solve the given equation, let's follow the steps below:

Step 1: Expand Both Sides of the Equation

First, we will expand both sides of the given equation:

Left Side:

x^2 - 7x x^2 - 7x

Right Side:

2x^2 - 8x - x - 4 2x^2 - 7x - 4

So the equation becomes:

x^2 - 7x 2x^2 - 7x - 4

Step 2: Rearrange the Equation

Next, we need to move all terms to one side of the equation to simplify:

x^2 - 7x - 2x^2 7x 4 0

Simplifying this, we get:

-x^2 4 0 or x^2 - 4 0

Step 3: Solve for x

Now, we can solve for x by taking the square root of both sides:

x^2 4

Taking the square root of both sides, we have:

x ±2

So, the solutions to the equation are:

x 2 and x -2

Verification

To ensure our solutions are correct, we can substitute x 2 and x -2 back into the original equation:

For x 2:

x^2 - 7x 2(2^2) - 1(2) - 4

4 - 14 8 - 4 - 4

-10 -10

For x -2:

x^2 - 7x 2(-2)^2 - 1(-2) - 4

4 14 8 2 - 4

18 18

Both substitutions confirm that our solutions are correct.

Additional Notes on Quadratic Equations

A quadratic equation can also be expressed in the form ax^2 bx c 0. In the case of the given equation, we arrived at x^2 - 4 0, which is a difference of squares. The general form for a difference of squares is a^2 - b^2 (a b)(a - b). Here, we can factor x^2 - 4 (x - 2)(x 2).

The solutions to the equation are the roots of the factors:

x - 2 0 or x 2 0

Solving these, we get:

x 2 and x -2

Summary

In conclusion, solving the given quadratic equation using factorization involves expanding, rearranging, simplifying, and solving for the variable. The solutions to the equation x^2 - 7x 2x^2 - 8x - x - 4 are x 2 and x -2. Verifying the solutions ensures accuracy.

Further Reading

For a deeper understanding of quadratic equations and factorization, consider exploring additional resources such as textbooks, online courses, or academic papers. Understanding these concepts will help in solving more complex algebraic problems.