Solving $x^2 - 8x - 9 = 0$ A Step-by-Step Guide

Introduction

In the realm of mathematics, quadratic equations hold a fundamental position, appearing in various fields, from physics to engineering. Understanding how to solve them is crucial for anyone delving into these disciplines. This article aims to provide a comprehensive guide on solving the quadratic equation $x^2 - 8x - 9 = 0$, walking you through each step with clarity and precision. We'll cover rewriting the equation, completing the square, applying the square root property, and ultimately finding the solutions for x. Whether you're a student grappling with algebra or a professional seeking a refresher, this guide will equip you with the knowledge and skills to tackle quadratic equations effectively.

Rewriting the Equation

To solve the quadratic equation $x^2 - 8x - 9 = 0$, our first step involves rewriting it into a more manageable form. Specifically, we want to express the equation in the form $x^2 + bx = c$. This transformation sets the stage for completing the square, a powerful technique for solving quadratic equations. To achieve this, we focus on isolating the terms containing x on one side of the equation and moving the constant term to the other side. In our case, we have $x^2 - 8x - 9 = 0$. The term we need to move is -9. To do this, we add 9 to both sides of the equation. This maintains the balance of the equation while shifting the constant term to the right side. Adding 9 to both sides, we get:

$x^2 - 8x - 9 + 9 = 0 + 9$

Simplifying this, we arrive at:

$x^2 - 8x = 9$

This is now in the desired form of $x^2 + bx = c$, where b is -8 and c is 9. Rewriting the equation in this manner is a crucial initial step because it prepares us for the next stage: completing the square. By isolating the x terms, we can focus on manipulating them to form a perfect square trinomial, which will greatly simplify the process of finding the solutions for x. This step highlights the importance of algebraic manipulation in solving equations and sets the foundation for more advanced techniques.

Completing the Square

Having rewritten the equation as $x^2 - 8x = 9$, the next critical step is to complete the square. Completing the square is a technique used to convert a quadratic expression into a perfect square trinomial, which is an expression that can be factored into the form $(x + a)^2$ or $(x - a)^2$. This method is particularly useful for solving quadratic equations that are not easily factorable. To complete the square for the expression $x^2 - 8x$, we need to add a constant term to both sides of the equation that will make the left side a perfect square. The constant term we need to add is determined by taking half of the coefficient of the x term and squaring it. In our equation, the coefficient of the x term is -8. Half of -8 is -4, and squaring -4 gives us 16. Therefore, we need to add 16 to both sides of the equation to complete the square. Adding 16 to both sides, we get:

$x^2 - 8x + 16 = 9 + 16$

This step is crucial because it transforms the left side of the equation into a perfect square trinomial. The expression $x^2 - 8x + 16$ can be factored as $(x - 4)^2$. On the right side, we simply add 9 and 16, which gives us 25. So, our equation now becomes:

$(x - 4)^2 = 25$

By completing the square, we have successfully converted the quadratic equation into a form that is much easier to solve. The perfect square trinomial allows us to apply the square root property in the next step, which will lead us to the solutions for x. This technique demonstrates the power of algebraic manipulation in simplifying complex equations and making them solvable.

Applying the Square Root Property

Now that we have the equation in the form $(x - 4)^2 = 25$, we can apply the square root property to find the values of x. The square root property states that if $a^2 = b$, then $a = ±√b$. In our case, $(x - 4)^2 = 25$, so we can take the square root of both sides of the equation. When taking the square root, it's crucial to remember that there are two possible solutions: a positive square root and a negative square root. Applying the square root property, we get:

$√(x - 4)^2 = ±√25$

This simplifies to:

$x - 4 = ±5$

Now we have two separate equations to solve:

1. $x - 4 = 5$
2. $x - 4 = -5$

Solving the first equation, $x - 4 = 5$, we add 4 to both sides:

$x = 5 + 4$

$x = 9$

Solving the second equation, $x - 4 = -5$, we also add 4 to both sides:

$x = -5 + 4$

$x = -1$

Thus, we have found two solutions for x: 9 and -1. The square root property is a powerful tool for solving equations in this form, as it allows us to eliminate the square and isolate the variable. This step highlights the importance of considering both positive and negative roots when solving equations, as neglecting one can lead to incomplete solutions. The solutions we have found are the values of x that satisfy the original quadratic equation.

Solutions and Verification

After applying the square root property, we have found two potential solutions for the quadratic equation $x^2 - 8x - 9 = 0$: x = 9 and x = -1. To ensure that these solutions are correct, it's essential to verify them by substituting each value back into the original equation. This process confirms that the solutions satisfy the equation and that no errors were made during the solving process. First, let's verify x = 9:

Substituting x = 9 into the original equation, we get:

$(9)^2 - 8(9) - 9 = 0$

$81 - 72 - 9 = 0$

$0 = 0$

Since the equation holds true, x = 9 is indeed a valid solution. Now, let's verify x = -1:

Substituting x = -1 into the original equation, we get:

$(-1)^2 - 8(-1) - 9 = 0$

$1 + 8 - 9 = 0$

$0 = 0$

Again, the equation holds true, confirming that x = -1 is also a valid solution. Therefore, the solutions to the quadratic equation $x^2 - 8x - 9 = 0$ are x = 9 and x = -1. This verification step is a crucial part of the problem-solving process, as it provides confidence in the accuracy of the solutions and helps to identify any potential errors. The solutions we have found represent the points where the parabola represented by the quadratic equation intersects the x-axis.

Conclusion

In this comprehensive guide, we have meticulously solved the quadratic equation $x^2 - 8x - 9 = 0$, demonstrating a step-by-step approach that can be applied to various quadratic equations. We began by rewriting the equation in the form $x^2 + bx = c$, which prepared us for the subsequent steps. Then, we employed the technique of completing the square, transforming the equation into a form that allowed us to easily apply the square root property. By applying the square root property, we found two potential solutions for x: 9 and -1. Finally, we verified these solutions by substituting them back into the original equation, confirming their validity. This process highlights the importance of algebraic manipulation, the significance of remembering both positive and negative roots, and the necessity of verifying solutions to ensure accuracy. Mastering these techniques is crucial for anyone working with quadratic equations, whether in academic or professional settings. Quadratic equations are a cornerstone of mathematics, and understanding how to solve them effectively opens doors to a deeper understanding of various mathematical and scientific concepts. This guide serves as a valuable resource for students, educators, and anyone seeking to enhance their problem-solving skills in mathematics.