Solve 2×(x-2)+8(x+1)-x(x-3)-(x²+7)= -2: Step-by-Step
Hey guys! Let's dive into solving this interesting algebraic equation: 2×(x-2)+8(x+1)-x(x-3)-(x²+7)= -2. Don't worry, we'll break it down step by step so it's super easy to follow. We'll cover each part in detail, ensuring you understand not just the how, but also the why behind each step. So grab your pencils and let's get started!
Understanding the Equation
Before we start crunching numbers, let's take a good look at the equation: 2×(x-2)+8(x+1)-x(x-3)-(x²+7)= -2. This equation looks a bit intimidating at first glance, but it's just a combination of terms involving 'x', constants, and some basic arithmetic operations. The key to solving it lies in simplifying each part, combining like terms, and then isolating 'x' on one side of the equation. It's like peeling an onion – layer by layer, we'll get to the core of the problem. We need to understand the order of operations (PEMDAS/BODMAS) to solve this accurately, which means Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Keep this in mind as we move forward; it’s our roadmap for simplifying the equation. Breaking down the equation into smaller, manageable parts will make the entire process less daunting. Trust me, by the end of this guide, you'll be able to tackle similar equations with confidence!
Step 1: Distribute and Expand
Okay, our first task is to get rid of those parentheses. We need to distribute the numbers outside the parentheses to the terms inside. Remember, distribution means multiplying the term outside the parentheses by each term inside. Let's start with the first part: 2×(x-2). We multiply 2 by both 'x' and '-2', which gives us 2x - 4. Next up is 8(x+1). Multiplying 8 by 'x' and '1' results in 8x + 8. Now for the trickier part: -x(x-3). Here, we're multiplying -x by both 'x' and '-3'. This gives us -x² + 3x. Finally, we have -(x²+7). This is like multiplying by -1, so we get -x² - 7. Now, let’s put it all together. Our equation now looks like this: 2x - 4 + 8x + 8 - x² + 3x - x² - 7 = -2. See? It’s already looking a bit more manageable. This step is crucial because it allows us to combine like terms in the next step. Without correctly distributing and expanding, we wouldn't be able to simplify the equation effectively. So, always double-check your distribution to avoid any silly mistakes!
Step 2: Combine Like Terms
Now that we've expanded the equation, it's time to combine like terms. This means grouping together terms that have the same variable and exponent (like x² terms, x terms, and constants). Let's start with the x² terms. We have -x² and -x², which combine to give us -2x². Next, let's look at the x terms: we have 2x, 8x, and 3x. Adding these together, we get 13x. Finally, let's combine the constants: -4, +8, and -7. This gives us -3. So, after combining like terms, our equation now looks like this: -2x² + 13x - 3 = -2. Doesn't that look much simpler? This step is really important because it reduces the number of terms in the equation, making it easier to solve. It's like tidying up a messy room – once everything is in its place, it’s much easier to find what you're looking for. Make sure you're careful when combining terms, paying close attention to the signs (positive or negative) in front of each term. A small mistake here can throw off the entire solution, so take your time and double-check your work!
Step 3: Move All Terms to One Side
To solve this quadratic equation, we need to set it equal to zero. This means moving all the terms to one side of the equation. We currently have -2x² + 13x - 3 = -2. To get rid of the -2 on the right side, we add 2 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. Adding 2 to both sides gives us: -2x² + 13x - 3 + 2 = -2 + 2. This simplifies to -2x² + 13x - 1 = 0. Great! Now our equation is in the standard quadratic form, which is ax² + bx + c = 0. This is a crucial step because it allows us to use methods like factoring, completing the square, or the quadratic formula to find the solutions for 'x'. By setting the equation to zero, we create a scenario where we can identify the values of a, b, and c, which are necessary for these solution methods. Always aim to get your equation into this form before attempting to solve for the variable. It’s like preparing your ingredients before you start cooking – it makes the whole process much smoother!
Step 4: Solve the Quadratic Equation
Now we have the quadratic equation -2x² + 13x - 1 = 0. There are a few ways to solve this, but since it doesn't look easily factorable, we'll use the quadratic formula. The quadratic formula is a powerful tool that can solve any quadratic equation. It’s given by: x = [-b ± √(b² - 4ac)] / (2a). In our equation, a = -2, b = 13, and c = -1. Let's plug these values into the formula. First, we calculate the discriminant, which is the part under the square root: b² - 4ac = 13² - 4(-2)(-1) = 169 - 8 = 161. Now we can plug everything into the quadratic formula: x = [-13 ± √161] / (2 * -2). This simplifies to x = [-13 ± √161] / -4. So, we have two possible solutions for x: x = (-13 + √161) / -4 and x = (-13 - √161) / -4. We can simplify these further by dividing both the numerator and the denominator by -1: x = (13 - √161) / 4 and x = (13 + √161) / 4. These are our exact solutions. If we need decimal approximations, we can use a calculator to find the square root of 161 (which is approximately 12.69) and then perform the calculations. This step is where all our hard work pays off. The quadratic formula might seem intimidating at first, but once you understand how to use it, it becomes a reliable method for solving quadratic equations. Remember to carefully substitute the values of a, b, and c, and double-check your calculations to avoid errors. Practice makes perfect, so the more you use the formula, the more comfortable you'll become with it!
Step 5: Approximate the Solutions (Optional)
Okay, we've found the exact solutions: x = (13 - √161) / 4 and x = (13 + √161) / 4. But sometimes, it's helpful to have decimal approximations to better understand the values. Let's use a calculator to approximate √161. We find that √161 is approximately 12.69. Now we can plug this into our solutions. For the first solution, x = (13 - 12.69) / 4, which is approximately 0.31 / 4, which is about 0.0775. For the second solution, x = (13 + 12.69) / 4, which is approximately 25.69 / 4, which is about 6.4225. So, our approximate solutions are x ≈ 0.0775 and x ≈ 6.4225. Having these approximations gives us a better sense of where the solutions lie on the number line. It's like having a map with estimated distances – it helps you visualize the journey better. While exact solutions are always preferred for accuracy, approximations can be useful in real-world applications or when comparing solutions. This step is optional, but it can provide valuable insights and context to your answers. Plus, it’s a good way to check if your exact solutions make sense in the context of the problem.
Conclusion
Alright, guys, we did it! We successfully solved the equation 2×(x-2)+8(x+1)-x(x-3)-(x²+7)= -2. We started by understanding the equation, then distributed and expanded, combined like terms, moved all terms to one side, solved the quadratic equation using the quadratic formula, and finally, approximated the solutions. Solving complex equations like this might seem daunting at first, but breaking it down into smaller, manageable steps makes the whole process much easier. Remember, the key is to take your time, be careful with your calculations, and double-check your work. Math is like a puzzle, and each step is a piece that fits together to reveal the solution. Practice is crucial – the more equations you solve, the more confident and skilled you'll become. So keep practicing, and you'll be a math whiz in no time! And remember, if you get stuck, don’t hesitate to review the steps or ask for help. Happy solving!
