Solving Quadratic Equations: A Step-by-Step Guide

Quadratic equations are a fundamental part of algebra, often appearing in various real-world applications and academic challenges. This article aims to provide a comprehensive guide on solving the specific equation x^2 - x 20 through a series of detailed steps, including expansion, rearrangement, factorization, and solution.

Understanding the Problem

The given equation is x^2 - x 20. Our goal is to determine the values of x that satisfy this equation, effectively solving for x.

Step-by-Step Solution

1. Expand and Rearrange the Equation

First, we start by moving the constant term to the other side to form a standard quadratic equation:

x^2 - x - 20 0

2. Factorize the Quadratic Equation

To factorize the quadratic equation, we need to find two numbers that multiply to give the constant term (-20) and add up to the coefficient of the middle term (-1).

The numbers that satisfy these conditions are -5 and 4, as:

-5 * 4 -20 -5 4 -1

Using the numbers -5 and 4, we can rewrite the equation as:

(x - 5)(x 4) 0

3. Solve for x

Now, we solve each factor separately to find the values of x:

For the factor x - 5 0, we get: x 5 For the factor x 4 0, we get: x -4

Conclusion

The solutions to the equation x^2 - x 20 are:

x 5 x -4

Additional Insights

Understanding the process of solving quadratic equations like x^2 - x 20 is essential in various fields, including engineering, physics, and economics. Quadratic equations with real roots have applications in determining areas, motion, and optimization problems.

This step-by-step approach can be applied to similar equations, making it a valuable skill for anyone working with algebraic equations. By mastering this method, you can confidently solve a wide range of algebraic problems involving quadratic relationships.

Keywords

Quadratic equation Solving equations Algebraic equations