Solving X² - 6x = -8 A Step-by-Step Guide

Jul 16, 2025 by ADMIN 42 views

Solving the Quadratic Equation x² - 6x = -8: A Comprehensive Guide

Hey guys! Today, we're diving into the world of quadratic equations and tackling the problem x² - 6x = -8. Don't worry, it's not as scary as it looks! We'll break it down step-by-step, so you'll be solving these like a pro in no time. Whether you're brushing up on your algebra skills or facing this problem in your homework, this guide is here to help. Quadratic equations are fundamental in mathematics and have numerous applications in various fields, including physics, engineering, and economics. Mastering the techniques to solve them is crucial for a strong foundation in algebra and beyond. This article aims to provide a comprehensive understanding of how to solve the quadratic equation x² - 6x = -8, covering different methods and offering clear, step-by-step explanations. By the end of this guide, you’ll not only know how to solve this specific equation but also understand the underlying principles that apply to a wide range of quadratic equations. So, let’s put on our math hats and get started!

Understanding Quadratic Equations

Before we jump into solving, let's make sure we're all on the same page about what a quadratic equation actually is. Quadratic equations, in their simplest form, are polynomial equations of the second degree. This means the highest power of the variable (usually 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. If 'a' were zero, the equation would become linear, not quadratic. Understanding this standard form is crucial because it provides a framework for applying various solution methods. The coefficients 'a', 'b', and 'c' play significant roles in determining the nature and number of solutions to the equation. For example, the sign and magnitude of 'b' influence the symmetry and position of the parabola represented by the quadratic equation when graphed, while 'c' determines the y-intercept. Recognizing the standard form also helps in identifying the appropriate method for solving the equation. Some equations are easily factorable, while others may require the use of the quadratic formula or completing the square. Each method has its advantages and disadvantages, and knowing when to apply each one is a key skill in algebra. So, keep that standard form in mind as we move forward!

Methods to Solve Quadratic Equations

There are several ways to solve quadratic equations, each with its own strengths. We'll focus on three primary methods:

Factoring: Factoring involves breaking down the quadratic expression into a product of two binomials. This method is efficient when the equation can be factored easily. Look for two numbers that multiply to 'c' and add up to 'b'. If you can find them, you're in business! Factoring is often the quickest method when it works, as it avoids complex calculations. However, not all quadratic equations are easily factorable, which is where the other methods come into play. The key to successful factoring is recognizing patterns and practicing different types of factorable expressions. For example, perfect square trinomials and the difference of squares are common patterns that can be factored readily. Additionally, understanding how to factor by grouping can be helpful for more complex quadratic expressions. While factoring is a valuable skill, it’s important to recognize its limitations and be prepared to use alternative methods when necessary.
Completing the Square: Completing the square is a powerful technique that transforms the quadratic equation into a perfect square trinomial. This method is particularly useful when the equation is not easily factorable. By adding and subtracting a specific value, we can rewrite the equation in a form that allows us to take the square root of both sides and solve for 'x'. Completing the square is not only a method for solving quadratic equations but also a fundamental concept in algebra. It demonstrates how to manipulate equations to reveal their underlying structure and is a building block for more advanced mathematical concepts. The process involves several steps, including ensuring the coefficient of x² is 1, adding and subtracting the square of half the coefficient of x, and then rewriting the equation in a perfect square form. While it may seem complex at first, mastering completing the square provides a deep understanding of quadratic equations and their properties. This method is especially useful when deriving the quadratic formula itself.
Quadratic Formula: The quadratic formula is the ultimate tool in our arsenal. It works for any quadratic equation, no matter how messy it looks. The formula is: x = [-b ± √(b² - 4ac)] / 2a. Just plug in the values of 'a', 'b', and 'c', and you'll get the solutions! The quadratic formula is derived from the method of completing the square, and it provides a direct way to find the solutions of any quadratic equation. It is a powerful and versatile tool, but it also requires careful attention to detail to avoid errors in calculation. The formula involves several operations, including squaring, subtraction, square root, and division, so it’s crucial to follow the order of operations correctly. Additionally, the expression inside the square root, known as the discriminant (b² - 4ac), provides valuable information about the nature of the solutions. If the discriminant is positive, there are two distinct real solutions; if it’s zero, there is one real solution (a repeated root); and if it’s negative, there are two complex solutions. Understanding the discriminant can help you anticipate the type of solutions you’ll find and check your work. So, while the quadratic formula might seem intimidating at first, it’s a reliable method that ensures you can solve any quadratic equation.

Solving x² - 6x = -8 by Factoring

Let's tackle our equation, x² - 6x = -8, using the factoring method first. Remember, the first step is to set the equation to zero. To do this, we add 8 to both sides:

x² - 6x + 8 = 0

Now, we need to find two numbers that multiply to 8 and add up to -6. Think about the factors of 8: 1 and 8, 2 and 4. Since we need a negative sum, both numbers should be negative. -2 and -4 fit the bill! They multiply to 8 and add up to -6.

So, we can factor the quadratic expression as follows:

(x - 2)(x - 4) = 0

Now, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This means either (x - 2) = 0 or (x - 4) = 0.

Solving these two simple equations, we get:

x - 2 = 0 => x = 2
x - 4 = 0 => x = 4

So, the solutions to the equation x² - 6x = -8 are x = 2 and x = 4. Factoring worked beautifully in this case, providing a quick and straightforward way to find the solutions. This illustrates the power of factoring when the equation is easily factorable. However, it’s important to note that not all quadratic equations can be factored this easily, which is why having other methods in your toolkit is essential. But for this particular equation, factoring was the perfect choice!

Solving x² - 6x = -8 by Completing the Square

Now, let's solve x² - 6x = -8 using the completing the square method. This method is a bit more involved but is a valuable technique to learn. Again, the first step is to make sure our equation is in a suitable form. Luckily, we already have the x² and x terms on one side, so we can proceed directly to completing the square.

The coefficient of our x² term is already 1, which is perfect. Now, we need to add and subtract the square of half the coefficient of our x term. The coefficient of x is -6, so half of that is -3, and the square of -3 is 9. We add and subtract 9 on the left side of the equation:

x² - 6x + 9 - 9 = -8

Notice that we've added and subtracted the same value, so we haven't changed the equation. However, the first three terms now form a perfect square trinomial:

(x - 3)² - 9 = -8

Now, we add 9 to both sides to isolate the squared term:

(x - 3)² = 1

Next, we take the square root of both sides:

x - 3 = ±√1

x - 3 = ±1

Finally, we solve for x by adding 3 to both sides:

x = 3 ± 1

This gives us two solutions:

x = 3 + 1 = 4
x = 3 - 1 = 2

As you can see, we got the same solutions as we did with factoring (x = 2 and x = 4), but through a different method. Completing the square is particularly useful when the quadratic equation is not easily factorable, and it provides a solid foundation for understanding the quadratic formula itself. While it might seem a bit more complex than factoring in this case, mastering this technique is a valuable skill in algebra.

Solving x² - 6x = -8 Using the Quadratic Formula

Finally, let's use the quadratic formula to solve x² - 6x = -8. This method is the most versatile and works for any quadratic equation, regardless of whether it can be factored easily or not. Remember, the quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

First, we need to rewrite our equation in the standard form ax² + bx + c = 0. We already did this when we moved the -8 to the left side:

x² - 6x + 8 = 0

Now, we can identify the coefficients: a = 1, b = -6, and c = 8. Plug these values into the quadratic formula:

x = [-(-6) ± √((-6)² - 4 * 1 * 8)] / (2 * 1)

Simplify step-by-step:

x = [6 ± √(36 - 32)] / 2

x = [6 ± √4] / 2

x = [6 ± 2] / 2

Now, we have two possible solutions:

x = (6 + 2) / 2 = 8 / 2 = 4
x = (6 - 2) / 2 = 4 / 2 = 2

Once again, we arrive at the same solutions: x = 2 and x = 4. The quadratic formula provides a reliable method for solving any quadratic equation, even those that are difficult to factor or complete the square. While it might involve more calculations than factoring in some cases, it guarantees a solution and is an essential tool in your mathematical toolkit.

Conclusion

Great job, guys! We've successfully solved the quadratic equation x² - 6x = -8 using three different methods: factoring, completing the square, and the quadratic formula. We found that the solutions are x = 2 and x = 4. Each method has its own advantages, and understanding all three gives you a powerful toolkit for tackling quadratic equations. Factoring is quick and efficient when it works, completing the square provides a deeper understanding of the equation's structure, and the quadratic formula is a reliable solution for any quadratic equation.

Remember, practice makes perfect! The more you work with these methods, the more comfortable and confident you'll become in solving quadratic equations. So, keep practicing, and you'll be a quadratic equation master in no time! Whether you're facing a similar problem in your homework or exploring more advanced mathematical concepts, the skills you’ve gained here will be invaluable. Quadratic equations are a fundamental part of algebra, and mastering them opens doors to more complex mathematical ideas. Keep exploring, keep learning, and most importantly, keep having fun with math!