Solving The Equation X² + 9x - 22 = 0 A Comprehensive Guide

by ADMIN 60 views

In the realm of mathematics, solving equations is a fundamental skill. One common type of equation encountered is the quadratic equation, which takes the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations pop up in various fields, from physics and engineering to economics and computer science. This article delves into the process of solving the specific quadratic equation x² + 9x - 22 = 0, providing a comprehensive understanding of the methods involved and the underlying principles.

Understanding Quadratic Equations

Before diving into the solution, let's grasp the core concepts of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this case, x) is 2. The coefficients a, b, and c determine the shape and position of the parabola that represents the equation when graphed. The solutions to the quadratic equation, also known as the roots or zeros, are the x-values where the parabola intersects the x-axis. These solutions represent the values of x that satisfy the equation, making the expression equal to zero.

Quadratic equations can have two distinct real roots, one repeated real root, or two complex roots. The nature of the roots depends on the discriminant (Δ), which is given by the formula Δ = b² - 4ac. If Δ > 0, there are two distinct real roots; if Δ = 0, there is one repeated real root; and if Δ < 0, there are two complex roots. In our case, a = 1, b = 9, and c = -22, so the discriminant is Δ = 9² - 4 * 1 * (-22) = 81 + 88 = 169. Since Δ > 0, we expect two distinct real roots.

Methods for Solving Quadratic Equations

There are several methods to tackle quadratic equations, each with its strengths and weaknesses. The most common methods are:

  1. Factoring: This method involves expressing the quadratic expression as a product of two linear factors. If the equation can be factored easily, this is often the quickest approach.
  2. Completing the Square: This method transforms the quadratic equation into a perfect square trinomial, allowing us to isolate the variable and solve for its values. It's a powerful technique that can be used even when factoring is difficult.
  3. Quadratic Formula: This formula provides a direct solution for any quadratic equation, regardless of whether it can be factored or not. It's a reliable method that always works, but it can be a bit more computationally intensive.

We'll explore each of these methods in the context of our equation, x² + 9x - 22 = 0.

Method 1: Factoring

Factoring is often the first method to try when solving a quadratic equation. The idea is to rewrite the quadratic expression as a product of two binomials. For the equation x² + 9x - 22 = 0, we need to find two numbers that multiply to -22 (the constant term) and add up to 9 (the coefficient of the x term). These numbers are 11 and -2, since 11 * (-2) = -22 and 11 + (-2) = 9.

Therefore, we can factor the quadratic expression as follows:

x² + 9x - 22 = (x + 11)(x - 2)

Now, to solve the equation, we set each factor equal to zero:

x + 11 = 0 or x - 2 = 0

Solving these linear equations, we get:

x = -11 or x = 2

Thus, the solutions to the equation x² + 9x - 22 = 0 are x = -11 and x = 2. Factoring proved to be an efficient method in this case, as the numbers involved were relatively straightforward.

Method 2: Completing the Square

Completing the square is another powerful technique for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial on one side. This method is particularly useful when the quadratic expression is not easily factorable. Let's apply this method to the equation x² + 9x - 22 = 0.

First, we move the constant term to the right side of the equation:

x² + 9x = 22

Next, we need to add a constant to both sides to complete the square on the left side. To find this constant, we take half of the coefficient of the x term (which is 9), square it ( (9/2)² = 81/4), and add it to both sides:

x² + 9x + 81/4 = 22 + 81/4

Now, the left side is a perfect square trinomial, which can be written as:

(x + 9/2)² = 22 + 81/4

To simplify the right side, we find a common denominator:

(x + 9/2)² = 88/4 + 81/4

(x + 9/2)² = 169/4

Now, we take the square root of both sides:

x + 9/2 = ±√(169/4)

x + 9/2 = ±13/2

We now have two separate equations to solve:

  1. x + 9/2 = 13/2 => x = 13/2 - 9/2 => x = 4/2 => x = 2
  2. x + 9/2 = -13/2 => x = -13/2 - 9/2 => x = -22/2 => x = -11

Again, we find the solutions x = 2 and x = -11, consistent with the factoring method. Completing the square, while a bit more involved, provides a systematic way to solve the equation.

Method 3: Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations. It provides a direct solution for any equation in the form ax² + bx + c = 0. The formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

For our equation, x² + 9x - 22 = 0, we have a = 1, b = 9, and c = -22. Plugging these values into the quadratic formula, we get:

x = (-9 ± √(9² - 4 * 1 * (-22))) / (2 * 1)

x = (-9 ± √(81 + 88)) / 2

x = (-9 ± √169) / 2

x = (-9 ± 13) / 2

This gives us two solutions:

  1. x = (-9 + 13) / 2 => x = 4 / 2 => x = 2
  2. x = (-9 - 13) / 2 => x = -22 / 2 => x = -11

As expected, we obtain the same solutions, x = 2 and x = -11. The quadratic formula is a reliable method that works for all quadratic equations, even those that are difficult or impossible to factor.

Verifying the Solutions

To ensure the accuracy of our solutions, we can substitute them back into the original equation, x² + 9x - 22 = 0.

For x = 2:

(2)² + 9(2) - 22 = 4 + 18 - 22 = 0

The equation holds true for x = 2.

For x = -11:

(-11)² + 9(-11) - 22 = 121 - 99 - 22 = 0

The equation also holds true for x = -11.

Therefore, our solutions x = 2 and x = -11 are correct.

Conclusion

In this article, we successfully solved the quadratic equation x² + 9x - 22 = 0 using three different methods: factoring, completing the square, and the quadratic formula. All three methods yielded the same solutions, x = 2 and x = -11. We also verified these solutions by substituting them back into the original equation. This exercise highlights the versatility of mathematical tools and the importance of understanding various problem-solving techniques. Mastering these methods equips you with the skills to tackle a wide range of mathematical challenges and real-world applications involving quadratic equations. Whether you prefer the elegance of factoring, the systematic approach of completing the square, or the reliability of the quadratic formula, the key is to choose the method that best suits your needs and the specific characteristics of the equation at hand. By understanding the underlying principles and practicing these techniques, you can confidently solve quadratic equations and unlock their potential in various fields of study and practical applications.

Remember, quadratic equations are a cornerstone of algebra and have far-reaching applications. From modeling projectile motion in physics to optimizing resource allocation in economics, the ability to solve these equations is a valuable skill. So, continue to explore and practice, and you'll find that the world of mathematics becomes increasingly accessible and rewarding.