Solving The Equation 18x - 5 = 6x(x + 2) + (x + 3) A Step-by-Step Guide

Hey there, math enthusiasts! Ever stumbled upon an equation that looks like a tangled mess? Well, you're not alone. Today, we're diving deep into the fascinating world of algebra to tackle the equation 18x - 5 = 6x(x + 2) + (x + 3). Don't worry, we'll break it down step by step, making sure everyone, from math newbies to seasoned pros, can follow along. So, buckle up, grab your thinking caps, and let's get started on this mathematical adventure!

Conquering Complex Equations: A Step-by-Step Guide

1. Simplifying the Equation: Taming the Algebraic Beast

Our journey begins with simplification. The key here is to get rid of those pesky parentheses and combine like terms. Think of it as decluttering your room – the cleaner it is, the easier it is to find what you need. When first encountering a complex equation like 18x - 5 = 6x(x + 2) + (x + 3), the initial reaction might be one of slight intimidation. However, the beauty of mathematics lies in its structured approach to problem-solving. The first crucial step in deciphering this equation is simplification. This process involves eliminating parentheses and grouping similar terms, akin to organizing a cluttered space to reveal its underlying order. So, let's roll our sleeves up and get into the nitty-gritty of algebraic manipulation!

Let's start by tackling the right side of the equation. We've got 6x(x + 2) + (x + 3). Remember the distributive property? It's our best friend here. We need to multiply 6x by both x and 2. This gives us 6x² + 12x. Now, we simply add the remaining term (x + 3). So, our right side becomes 6x² + 12x + x + 3. Now, before we move on, let's talk about the importance of understanding the distributive property. It's one of the fundamental principles in algebra, and mastering it is crucial for tackling more complex equations. The distributive property, in essence, states that for any numbers a, b, and c, a(b + c) = ab + ac. In simpler terms, it means that you can multiply a single term by a group of terms inside parentheses by multiplying it individually with each term within the group. This principle is not just a mathematical trick; it's a logical tool that allows us to break down complex expressions into manageable parts. Think of it like this: you have a package to deliver to a group of houses on a street. Instead of delivering the package to the entire street at once (which is impossible), you deliver it to each house individually. The distributive property works in the same way, allowing us to distribute the multiplication over addition or subtraction. Now, back to our equation. We've applied the distributive property to expand the expression 6x(x + 2), resulting in 6x² + 12x. This expansion is a critical step because it transforms the equation from a form that's difficult to work with into one that's more amenable to algebraic manipulation. By understanding and applying the distributive property, we've taken the first step in simplifying our equation, paving the way for further steps in the problem-solving process. Remember, in mathematics, each step builds upon the previous one. So, a solid understanding of the basics, like the distributive property, is essential for tackling more complex challenges.

But wait, we're not done yet! We can combine the 12x and x terms to get 13x. So, the right side is now 6x² + 13x + 3. Our equation looks much cleaner now: 18x - 5 = 6x² + 13x + 3. By simplifying the right side of the equation, we've made it easier to compare with the left side and identify like terms. This is a crucial step in solving any algebraic equation because it helps us consolidate the information and see the underlying structure more clearly. Combining like terms is like sorting your socks – you group the matching pairs together to make them easier to manage. In our equation, 12x and x are like terms because they both contain the variable x raised to the power of 1. We can combine them by adding their coefficients (the numbers in front of the x), which in this case are 12 and 1. So, 12x + x becomes 13x. This simple addition has a significant impact on the overall equation. It reduces the number of terms we need to deal with, making the equation less cluttered and more approachable. Think of it as condensing a long paragraph into a concise summary – you retain the essential information while eliminating unnecessary details. The process of combining like terms is not just about simplifying the equation; it's also about revealing the underlying relationships between the different parts. By grouping similar terms together, we can see how they interact and influence each other. This understanding is crucial for making informed decisions about the next steps in the problem-solving process. In our equation, we've combined the x terms, but we might also have constant terms (numbers without variables) or terms with higher powers of x (like x²) that need to be combined. The principle remains the same: identify the like terms, add or subtract their coefficients, and simplify the equation. Remember, simplification is not just about making the equation look neater; it's about making it easier to solve. By combining like terms, we've taken a significant step towards unraveling the mystery of our equation and finding its solutions.

2. Rearranging the Equation: Setting the Stage for Solutions

Now, let's rearrange the equation to get all the terms on one side, leaving zero on the other. This sets the stage for solving the equation. We want to get our equation into the standard quadratic form, which is ax² + bx + c = 0. The transition from a simplified equation to the standard quadratic form is a pivotal moment in our problem-solving journey. It's like arranging your chess pieces before making your first move – you need to position them strategically to maximize your chances of success. The standard quadratic form, ax² + bx + c = 0, is not just a convention; it's a powerful tool that allows us to apply various techniques for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. By rearranging our equation into this form, we're essentially unlocking a treasure chest of mathematical methods that can help us find the solutions. So, how do we go about rearranging our equation? The key is to use inverse operations to move terms from one side of the equation to the other. Remember, the golden rule of algebra is that whatever you do to one side of the equation, you must do to the other side to maintain the balance. In our case, we want to get all the terms on one side, leaving zero on the other. This means we need to eliminate the terms on the left side of the equation, 18x - 5. To eliminate 18x, we subtract 18x from both sides of the equation. This gives us -5 = 6x² + 13x + 3 - 18x. Similarly, to eliminate -5, we add 5 to both sides of the equation. This results in 0 = 6x² + 13x + 3 - 18x + 5. Now, we need to simplify the right side by combining like terms. We have 13x and -18x, which combine to give us -5x. We also have the constant terms 3 and 5, which combine to give us 8. So, our equation in the standard quadratic form is 0 = 6x² - 5x + 8. This transformation is a significant achievement. We've taken our equation from a jumble of terms and rearranged it into a structured form that's ready for the next stage of the solution process. By understanding the principles of inverse operations and the importance of maintaining balance in the equation, we've successfully navigated this crucial step. Remember, rearranging the equation is not just about following a set of rules; it's about strategically positioning ourselves to solve the problem effectively. The standard quadratic form is our chessboard, and the techniques we'll use next are our chess moves. So, let's prepare ourselves for the next stage and get ready to checkmate this equation!

Let's subtract 18x from both sides: -5 = 6x² + 13x - 18x + 3. And then, add 5 to both sides: 0 = 6x² + 13x - 18x + 3 + 5. Now, combine those like terms again! We get 0 = 6x² - 5x + 8. Awesome, we're in the standard quadratic form now!

3. Solving the Quadratic Equation: Cracking the Code

Now comes the exciting part – finding the solutions! We have a few options here. We could try factoring, but sometimes that's tricky. Another option is using the quadratic formula, which always works. The climax of our mathematical quest arrives as we focus on solving the quadratic equation. We've meticulously simplified, rearranged, and positioned our equation into the standard form, ax² + bx + c = 0, which is 0 = 6x² - 5x + 8. Now, we're ready to deploy the tools and techniques that will reveal the values of x that satisfy this equation. It's like reaching the final level of a video game, where the ultimate challenge awaits, but also the ultimate reward of victory. When it comes to solving quadratic equations, we have a few powerful options in our arsenal. One approach is factoring, where we try to express the quadratic expression as a product of two binomials. Factoring is like unlocking a secret code that reveals the hidden structure of the equation. If we can factor the equation, we can easily find the solutions by setting each factor equal to zero and solving for x. However, factoring is not always straightforward. It requires a bit of trial and error, and sometimes the equation might not be factorable using simple methods. This is where the quadratic formula comes to our rescue. The quadratic formula is a universal tool that can solve any quadratic equation, regardless of whether it's factorable or not. It's like having a master key that can open any lock. The formula itself might look a bit intimidating at first, but it's actually quite straightforward to use once you understand its components. The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation in the standard form. In our equation, a = 6, b = -5, and c = 8. Plugging these values into the formula, we get: x = (5 ± √((-5)² - 4 * 6 * 8)) / (2 * 6). Now, we need to simplify the expression inside the square root. We have (-5)² = 25 and 4 * 6 * 8 = 192. So, the expression becomes 25 - 192 = -167. This is where things get interesting. We have a negative number inside the square root. In the realm of real numbers, the square root of a negative number is undefined. This means that our equation has no real solutions. However, in the realm of complex numbers, we can take the square root of a negative number. The square root of -1 is denoted by the imaginary unit i. So, the square root of -167 is √167 * i. Therefore, the solutions to our equation are complex numbers. This discovery adds another layer of depth to our problem-solving journey. We've not only found the solutions, but we've also uncovered the nature of those solutions – they're complex! By understanding the quadratic formula and the concept of complex numbers, we've successfully cracked the code of our quadratic equation. Remember, in mathematics, the journey is just as important as the destination. We've learned valuable techniques and concepts along the way, which will serve us well in future mathematical endeavors. So, let's celebrate our victory and move on to the next challenge!

The quadratic formula is: x = (-b ± √(b² - 4ac)) / 2a. Remember that? In our equation, a = 6, b = -5, and c = 8. Let's plug those values in!

So, we get x = (5 ± √((-5)² - 4 * 6 * 8)) / (2 * 6). Simplify that a bit, and we have x = (5 ± √(25 - 192)) / 12. Uh oh, we have a negative number under the square root! That means our solutions are complex numbers.

4. Delving into Complex Solutions: Embracing the Imaginary

Since we have a negative number under the square root, our solutions will involve imaginary numbers. Remember i, the imaginary unit, where i² = -1? This might sound intimidating, but it's actually a super cool part of math! The appearance of a negative number under the square root is not a roadblock; it's an invitation to explore the fascinating realm of complex numbers. It's like discovering a hidden passage in a castle that leads to a secret chamber filled with mathematical treasures. Complex numbers are not just abstract concepts; they have real-world applications in various fields, including electrical engineering, quantum mechanics, and fluid dynamics. So, by delving into complex solutions, we're expanding our mathematical horizons and gaining a deeper understanding of the world around us. When we encounter a negative number under the square root, we can't simply ignore it or say that there are no solutions. We need to embrace the imaginary unit i, which is defined as the square root of -1. This seemingly simple concept opens up a whole new dimension of numbers. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The real part of the complex number is a, and the imaginary part is b. In our case, we have √(25 - 192) = √(-167). We can rewrite this as √(167 * -1) = √167 * √-1 = √167 * i. So, the square root of -167 is a purely imaginary number, with a real part of 0 and an imaginary part of √167. Now, let's go back to our quadratic formula: x = (5 ± √(25 - 192)) / 12. We can substitute √167 * i for √(25 - 192), giving us: x = (5 ± √167 * i) / 12. This means we have two complex solutions: x = (5 + √167 * i) / 12 and x = (5 - √167 * i) / 12. These solutions are complex conjugates of each other, meaning they have the same real part but opposite imaginary parts. Complex conjugate pairs often arise as solutions to quadratic equations with real coefficients. By embracing the imaginary unit and understanding the concept of complex numbers, we've successfully navigated the challenge of negative numbers under the square root. We've not only found the solutions to our equation, but we've also gained a valuable insight into the broader landscape of mathematics. Remember, the beauty of mathematics lies in its ability to surprise and challenge us. By embracing the unexpected and exploring new concepts, we can unlock a deeper understanding of the world around us.

So, our solutions are x = (5 ± √(-167)) / 12. We can rewrite √(-167) as √167 * i. Therefore, our final solutions are x = (5 ± √167 * i) / 12. There you have it – complex solutions to a quadratic equation!

Conclusion: Math is an Adventure!

Wow, we've really tackled a complex equation today! We simplified, rearranged, and used the quadratic formula to find our solutions. And guess what? We even discovered complex numbers along the way! Remember, guys, math isn't just about numbers and formulas; it's about problem-solving and critical thinking. So, keep exploring, keep learning, and keep those mathematical gears turning! Solving an equation like 18x - 5 = 6x(x + 2) + (x + 3) is like embarking on a thrilling adventure. It's a journey filled with twists and turns, challenges and triumphs. We start with a seemingly complex problem, but through a systematic approach and the application of mathematical principles, we unravel its mysteries and arrive at the solutions. This journey is not just about finding the answers; it's about developing our problem-solving skills, enhancing our critical thinking abilities, and expanding our mathematical knowledge. Each step we take, from simplifying the equation to applying the quadratic formula, is a milestone in our adventure. We encounter challenges along the way, such as negative numbers under the square root, but we overcome them by embracing new concepts, like complex numbers. The discovery of complex solutions is like stumbling upon a hidden treasure – it adds another layer of richness and depth to our understanding of mathematics. Throughout this adventure, we learn the importance of perseverance, attention to detail, and a willingness to explore new ideas. We realize that mathematics is not just about memorizing formulas and applying procedures; it's about understanding the underlying concepts and using them creatively to solve problems. The quadratic formula, for example, is not just a formula to be memorized; it's a powerful tool that allows us to solve a wide range of quadratic equations. By understanding its derivation and application, we gain a deeper appreciation for its elegance and effectiveness. As we reach the end of our adventure, we not only have the solutions to our equation, but we also have a sense of accomplishment and a renewed enthusiasm for mathematics. We've proven to ourselves that we can tackle complex problems and that mathematics is not just a subject to be studied, but a tool to be used and a world to be explored. So, let's continue our mathematical adventures with confidence and curiosity, knowing that each challenge we face is an opportunity to learn and grow. Remember, guys, math is not a spectator sport; it's an active exploration. So, grab your mathematical gear, put on your thinking caps, and let's embark on the next adventure!