Solving X² + X - 20 = 0 A Comprehensive Guide

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In the realm of mathematics, quadratic equations hold a significant place. They appear in various scientific and engineering applications, making it crucial to understand how to solve them effectively. This article delves into the process of finding the solutions, also known as roots, of a specific quadratic equation: x² + x - 20 = 0. We will explore different methods to solve this equation and verify the solutions provided in the options: A. -4, B. -20, C. 5, D. -5, and E. 4.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation are the values of x that satisfy the equation. These solutions are also known as roots or zeros of the equation. Finding the roots of a quadratic equation is a fundamental problem in algebra, and several methods exist to tackle it.

Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations, including:

  1. Factoring: This method involves expressing the quadratic equation as a product of two linear factors. Factoring is an efficient method when the quadratic equation can be easily factored.
  2. Completing the Square: This method involves manipulating the quadratic equation to form a perfect square trinomial. Completing the square is a versatile method that can be used to solve any quadratic equation.
  3. Quadratic Formula: The quadratic formula is a general formula that provides the solutions to any quadratic equation. It is derived from the method of completing the square and is a reliable method for solving quadratic equations.

In this article, we will primarily use the factoring method and the quadratic formula to find the solutions to the given equation.

Solving x² + x - 20 = 0 by Factoring

Factoring is a method of expressing a quadratic expression as the product of two linear expressions. This method is efficient when the quadratic expression can be easily factored. To solve the equation x² + x - 20 = 0 by factoring, we need to find two numbers that multiply to -20 and add up to 1 (the coefficient of the x term).

Let's analyze the factors of -20:

  • 1 and -20
  • -1 and 20
  • 2 and -10
  • -2 and 10
  • 4 and -5
  • -4 and 5

Among these pairs, the pair -4 and 5 satisfy the condition that their product is -20 and their sum is 1. Therefore, we can factor the quadratic expression as follows:

x² + x - 20 = (x - 4)(x + 5)

Now, to solve the equation (x - 4)(x + 5) = 0, we set each factor equal to zero:

  • x - 4 = 0 => x = 4
  • x + 5 = 0 => x = -5

Thus, the solutions to the equation x² + x - 20 = 0 by factoring are x = 4 and x = -5.

Solving x² + x - 20 = 0 Using the Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. The formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

In our equation, x² + x - 20 = 0, we have a = 1, b = 1, and c = -20. Substituting these values into the quadratic formula, we get:

x = (-1 ± √(1² - 4(1)(-20))) / 2(1)

x = (-1 ± √(1 + 80)) / 2

x = (-1 ± √81) / 2

x = (-1 ± 9) / 2

This gives us two possible solutions:

  • x = (-1 + 9) / 2 = 8 / 2 = 4
  • x = (-1 - 9) / 2 = -10 / 2 = -5

Therefore, using the quadratic formula, the solutions to the equation x² + x - 20 = 0 are x = 4 and x = -5. This confirms the solutions we found using the factoring method.

Verifying the Solutions

Now that we have found the solutions, it's essential to verify them by substituting them back into the original equation x² + x - 20 = 0.

Verifying x = 4

Substituting x = 4 into the equation:

(4)² + 4 - 20 = 16 + 4 - 20 = 20 - 20 = 0

Since the equation holds true, x = 4 is a valid solution.

Verifying x = -5

Substituting x = -5 into the equation:

(-5)² + (-5) - 20 = 25 - 5 - 20 = 25 - 25 = 0

Since the equation holds true, x = -5 is also a valid solution.

Analyzing the Given Options

We are given the following options:

A. -4 B. -20 C. 5 D. -5 E. 4

Based on our calculations, the solutions to the equation x² + x - 20 = 0 are x = 4 and x = -5. Therefore, the correct options are:

  • D. -5
  • E. 4

Common Mistakes to Avoid

When solving quadratic equations, it's important to avoid common mistakes that can lead to incorrect solutions. Some of these mistakes include:

  1. Incorrect Factoring: Ensure that the factors you find correctly multiply to the original quadratic expression. Double-check your factors to avoid errors.
  2. Sign Errors: Pay close attention to the signs of the coefficients and constants in the equation. Sign errors can significantly affect the solutions.
  3. Misapplication of the Quadratic Formula: Ensure that you correctly substitute the values of a, b, and c into the quadratic formula. A slight mistake can lead to incorrect results.
  4. Forgetting to Verify Solutions: Always verify your solutions by substituting them back into the original equation. This will help you catch any errors and ensure that your solutions are correct.

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous applications in the real world. Some of these applications include:

  1. Physics: Quadratic equations are used to model projectile motion, the trajectory of objects thrown into the air.
  2. Engineering: They are used in the design of bridges, buildings, and other structures, as well as in electrical circuits and signal processing.
  3. Economics: Quadratic equations can model cost, revenue, and profit functions, helping businesses make informed decisions.
  4. Computer Graphics: They are used in creating curves and surfaces in computer graphics and animation.
  5. Finance: Quadratic equations can be used to model financial growth and investment returns.

The wide range of applications underscores the importance of understanding quadratic equations and how to solve them effectively.

Conclusion

In this article, we explored the process of solving the quadratic equation x² + x - 20 = 0. We used both the factoring method and the quadratic formula to find the solutions, which are x = 4 and x = -5. We also verified these solutions by substituting them back into the original equation. Furthermore, we discussed common mistakes to avoid when solving quadratic equations and highlighted the real-world applications of these equations.

Understanding how to solve quadratic equations is a fundamental skill in mathematics, and it opens the door to a wide range of applications in various fields. By mastering the methods discussed in this article, you can confidently tackle quadratic equations and apply them to solve real-world problems.