Solving Cubic Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of algebra and tackle a fun problem: finding all the solutions to the cubic equation x³ + 8x² + 100x = -800. Cubic equations might seem a bit intimidating at first, but trust me, with a systematic approach, we can break them down and find those elusive solutions. In this comprehensive guide, we'll walk through the process step-by-step, making sure you understand every concept along the way. Get ready to flex those math muscles! We'll explore techniques like rearranging the equation, factoring, and using the quadratic formula. By the end, you'll be able to confidently solve this type of equation. It's all about breaking down the problem into smaller, more manageable parts. So, grab your pencils, your calculators (if you want), and let's get started. Remember, practice makes perfect, so don't be afraid to try this problem on your own after we go through it together. Also, don't worry if you get stuck, that's part of the learning process. The key is to keep going and to keep practicing.

Before we begin, remember that a cubic equation is a polynomial equation of degree three. This means the highest power of the variable (in our case, x) is 3. Cubic equations can have up to three solutions (also known as roots), which can be real or complex numbers. Our goal is to find all of these solutions. So, are you ready to embark on this journey?

Step 1: Rearrange the Equation

The first and crucial step in solving this cubic equation is to rearrange it into a standard form. We want to have all the terms on one side of the equation and zero on the other side. This is like setting up the battlefield before the battle begins. So, let's take our equation: x³ + 8x² + 100x = -800. To get all the terms on one side, we simply add 800 to both sides: x³ + 8x² + 100x + 800 = 0. Now, we have our equation in standard form, ready for further analysis. This rearrangement is crucial because it allows us to utilize various techniques to solve the equation. It's like preparing the ingredients before starting a recipe.

Having the equation in this form makes it easier to spot patterns, apply factoring techniques, or use other methods to find the roots. The standard form provides a clear starting point, enabling us to systematically find the solutions. Keep in mind that getting the equation into this form is a fundamental step and the foundation for the rest of the process. It's a way of setting the stage and making sure everything is in place for the next steps. Now that we have our equation in standard form, we can move on to the next step, which will help us solve it.

This simple rearrangement has given us a clear starting point. We've ensured that the equation is in a format that's conducive to finding its solutions. Remember, attention to detail at this early stage sets the foundation for success. It is important to remember this, because any mistakes here will cause errors throughout the process. Now we are ready for the next step.

Step 2: Look for Patterns and Factoring

Once we have the equation in standard form (x³ + 8x² + 100x + 800 = 0), our next mission is to search for patterns and consider factoring. Factoring is a powerful technique that can simplify complex equations by breaking them down into smaller, more manageable parts. In simpler terms, we're trying to rewrite the equation as a product of factors, which will allow us to find the roots more easily. The ultimate goal is to try to rewrite the equation in a way that allows us to find the solutions. There are many methods for doing this, but the general principle is to look for common factors or recognizable patterns.

Let's examine our equation, x³ + 8x² + 100x + 800 = 0. Notice that the first two terms (x³ + 8x²) have a common factor of x². Factoring out x² from these terms, we get x²(x + 8). Now, let's look at the last two terms (100x + 800). They have a common factor of 100. Factoring out 100 from these terms, we get 100(x + 8). Now we can rewrite the equation in a somewhat factored form: x²(x + 8) + 100(x + 8) = 0.

This is where things get interesting! Observe that both terms now have a common factor of (x + 8). This is a very good sign because it opens up the possibility of further factoring. We can factor out (x + 8) from the entire expression, which gives us (x + 8)(x² + 100) = 0. Success! We've managed to factor the cubic equation into two factors: (x + 8) and (x² + 100).

Now, we have a product of two factors equal to zero. This is a critical point because it tells us that either the first factor is zero or the second factor is zero. This principle is fundamental in solving factored equations. It means that to find the solutions, we need to set each factor equal to zero and solve the resulting equations. This is a clever approach that simplifies our problem significantly. And also, that's exactly what we are going to do next!

Step 3: Solve for the Roots

We have successfully factored the cubic equation into (x + 8)(x² + 100) = 0. Now, it's time to solve for the roots. As we mentioned, if the product of two factors is zero, then at least one of the factors must be zero. Let's start with the first factor, (x + 8) = 0. To find the solution, we simply subtract 8 from both sides, which gives us x = -8. This is our first root, and it's a real number.

Now, let's consider the second factor, (x² + 100) = 0. This looks slightly more complex, but we can solve it by isolating x². Subtracting 100 from both sides, we get x² = -100. To solve for x, we need to take the square root of both sides. However, the square root of a negative number is not a real number. This indicates that the solutions will be complex numbers. Therefore, taking the square root of both sides gives us x = ±√(-100). We know that the square root of -1 is represented by i, the imaginary unit. So, √(-100) = √(-1) * √100 = 10i*. Therefore, the solutions for the second factor are x = 10i and x = -10i.

We have now found all three solutions to the cubic equation. One real root: x = -8, and two complex conjugate roots: x = 10i and x = -10i. These are all the values of x that satisfy the original equation. Each of these solutions, when substituted back into the original equation, will make the equation true. The process of finding these solutions, step-by-step, is the core of solving the cubic equation. It shows us how different mathematical techniques can be applied together to solve complex problems.

By following these steps, we've not only solved the cubic equation but also gained a better understanding of the techniques involved in solving such equations. Remember, the journey is just as important as the destination, and hopefully, you're now more confident when faced with similar challenges. Also, do not forget to practice, so you can master solving these types of equations. You got this!

Step 4: Verification and Conclusion

To ensure our solutions are correct, we should always verify them. Let's substitute each solution back into the original equation (x³ + 8x² + 100x = -800) to check if it holds true.

For x = -8: (-8)³ + 8(-8)² + 100(-8) = -512 + 512 - 800 = -800. This is true, so x = -8 is a valid solution.

For x = 10i: (10i)³ + 8(10i)² + 100(10i) = -1000i - 800 + 1000i = -800. This is also true.

For x = -10i: (-10i)³ + 8(-10i)² + 100(-10i) = 1000i - 800 - 1000i = -800. This is true as well.

We have verified that all three solutions satisfy the original equation. We've successfully found all the roots of the cubic equation: x = -8, x = 10i, and x = -10i. The solution x = -8 is a real root, while x = 10i and x = -10i are complex conjugate roots. These results show that a cubic equation can have both real and complex solutions, and our systematic approach allowed us to identify all of them.

We've covered all the necessary steps to solve this cubic equation, from rearranging it into standard form, to factoring, finding the roots, and verifying the solutions. The ability to solve cubic equations is an important skill in algebra, and it serves as a foundation for more advanced topics in mathematics. Keep practicing these techniques, and you'll find yourself getting better at solving these kinds of problems. Remember, the more you practice, the more familiar you'll become with the process. The process might seem difficult at first, but with persistence, you will be able to master it.

Congratulations on completing this problem! You can use these steps to solve similar cubic equations. Keep up the great work, and don't hesitate to explore more problems to hone your skills. The world of mathematics is full of fascinating challenges, and each problem you solve will take you one step closer to mastery. So, keep learning, keep practicing, and keep exploring! Now that you know how to solve this kind of equation, try to tackle some other complex problems! You are already on the right track!