Solving 2x² - 5x + 7 = X² - 2: A Step-by-Step Guide

Hey guys! Let's dive into the fascinating world of quadratic equations and tackle the solution set of the equation 2x² - 5x + 7 = x² - 2. Quadratic equations might seem intimidating at first, but with a systematic approach, we can crack them open and find their solutions. This comprehensive guide will walk you through each step, ensuring you understand the underlying concepts and can confidently solve similar problems in the future. So, grab your pencils and notebooks, and let's get started!

Understanding Quadratic Equations

Before we jump into solving our specific equation, it's crucial to have a solid grasp of what quadratic equations are all about. Quadratic equations are polynomial equations of the second degree, meaning 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 0. The solutions to a quadratic equation are also known as its roots or zeros, and they represent the values of 'x' that make the equation true. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its strengths and weaknesses, and the best approach often depends on the specific equation you're dealing with. Factoring is a great method when the quadratic expression can be easily factored into two linear expressions. Completing the square is a powerful technique that can be used to solve any quadratic equation, and it also provides a pathway to deriving the quadratic formula. And speaking of the quadratic formula, it's a versatile tool that guarantees a solution, even when factoring proves difficult or impossible. It's like having a Swiss Army knife for quadratic equations! Understanding these fundamental concepts is key to tackling the problem at hand and mastering quadratic equations in general. Remember, practice makes perfect, so the more you work with these equations, the more comfortable and confident you'll become. Let's move on to the first step in solving our equation: simplifying it.

Step 1: Simplifying the Equation

The first step in solving any equation, especially a quadratic one, is to simplify it. Simplifying makes the equation easier to work with and helps us identify the best method for finding the solutions. Our equation is 2x² - 5x + 7 = x² - 2. The goal here is to get all the terms on one side of the equation, leaving zero on the other side. This will bring our equation into the standard quadratic form (ax² + bx + c = 0). To do this, we'll subtract x² from both sides of the equation: 2x² - x² - 5x + 7 = x² - x² - 2, which simplifies to x² - 5x + 7 = -2. Next, we'll add 2 to both sides to get everything on the left side: x² - 5x + 7 + 2 = -2 + 2, which simplifies to x² - 5x + 9 = 0. Now, we have a simplified quadratic equation in the standard form. This form is crucial because it allows us to easily identify the coefficients 'a', 'b', and 'c', which we'll need for methods like the quadratic formula. In our simplified equation, x² - 5x + 9 = 0, we can see that a = 1, b = -5, and c = 9. With the equation simplified, we're now ready to choose the best method to solve it. In this case, factoring might be tricky, so let's consider other options like the quadratic formula or completing the square. But before we jump to those methods, it's always a good idea to check if factoring is possible, as it's often the quickest route to the solution. So, let's briefly explore factoring before moving on.

Step 2: Exploring Factoring (and Why It Might Not Work)

Factoring is often the first method we try when solving quadratic equations because it can be the quickest way to find the solutions. The idea behind factoring is to rewrite the quadratic expression as a product of two linear expressions. For example, if we could rewrite x² - 5x + 9 as (x - p)(x - q), then the solutions would simply be x = p and x = q. However, not all quadratic equations can be easily factored. To see if our equation, x² - 5x + 9 = 0, can be factored, we need to find two numbers that multiply to 'c' (which is 9) and add up to 'b' (which is -5). Let's think about the factors of 9: 1 and 9, or 3 and 3. Can we combine any of these factors to get -5? Well, -1 and -9 add up to -10, and -3 and -3 add up to -6. None of these combinations give us -5. This suggests that our quadratic equation might not factor neatly using integers. When factoring doesn't seem straightforward, it's a good indication that we should explore other methods, such as the quadratic formula or completing the square. These methods are more versatile and can handle quadratic equations that don't factor easily. In this case, since factoring isn't panning out, let's move on to the quadratic formula, which is a reliable tool for solving any quadratic equation. We'll use the coefficients we identified earlier (a = 1, b = -5, and c = 9) and plug them into the formula to find the solutions. So, let's get ready to unleash the power of the quadratic formula!

Step 3: Applying the Quadratic Formula

Alright guys, let's bring out the big guns – the quadratic formula! This formula is a lifesaver when factoring doesn't work, and it guarantees we'll find the solutions to any quadratic equation. The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / 2a. Remember those coefficients we identified earlier? (a = 1, b = -5, and c = 9). Now's the time to put them to work. Let's plug these values into the quadratic formula: x = (-(-5) ± √((-5)² - 4 * 1 * 9)) / (2 * 1). Now, let's simplify step by step. First, we have -(-5), which is simply 5. Next, let's look at the expression inside the square root: (-5)² is 25, and 4 * 1 * 9 is 36. So, we have 25 - 36, which equals -11. Our equation now looks like this: x = (5 ± √(-11)) / 2. Uh oh, we have a negative number inside the square root! This means our solutions will be complex numbers. Don't worry, this is perfectly normal in some quadratic equations. The square root of -11 can be written as √11 * √-1, and we know that √-1 is represented by the imaginary unit 'i'. So, √(-11) becomes i√11. Now, let's plug this back into our equation: x = (5 ± i√11) / 2. This gives us two complex solutions: x = (5 + i√11) / 2 and x = (5 - i√11) / 2. These are the solutions to our quadratic equation! We've successfully navigated the quadratic formula and found the complex roots. This highlights the power and versatility of the quadratic formula in solving quadratic equations, even those with complex solutions. Now that we have our solutions, let's summarize them and discuss what they mean in the context of the original equation.

Step 4: Interpreting the Solutions

Fantastic job, guys! We've successfully navigated the quadratic formula and arrived at our solutions: x = (5 + i√11) / 2 and x = (5 - i√11) / 2. But what do these solutions actually tell us? Well, these are the values of 'x' that make the equation 2x² - 5x + 7 = x² - 2 true. However, these solutions are complex numbers, meaning they have both a real part and an imaginary part (the part with 'i'). This has a significant implication: the graph of the quadratic equation y = x² - 5x + 9 (which is the simplified form of our original equation) does not intersect the x-axis. Why? Because the x-axis represents real numbers, and our solutions are complex. If the graph intersected the x-axis, we would have real solutions. Think of it this way: complex solutions indicate that the parabola represented by the quadratic equation is floating either entirely above or entirely below the x-axis. It never touches or crosses it. This is a crucial insight into the nature of quadratic equations and their graphical representations. When you encounter complex solutions, it's a signal that the parabola has no real roots. To further solidify our understanding, let's recap the entire process we followed to solve this equation. We started by simplifying the equation, then explored factoring (which didn't work in this case), and finally, we applied the quadratic formula to find the complex solutions. This systematic approach is key to tackling any quadratic equation, and understanding the nature of the solutions (real or complex) adds another layer of insight. So, pat yourselves on the back, guys! You've successfully tackled a quadratic equation with complex solutions. Now, let's move on to a summary of our findings and some final thoughts.

Conclusion: Solution Set and Key Takeaways

Alright guys, we've reached the end of our journey to find the solution set of the equation 2x² - 5x + 7 = x² - 2. To recap, we simplified the equation to x² - 5x + 9 = 0, explored factoring (which didn't work out this time), and then confidently applied the quadratic formula. This led us to the complex solutions: x = (5 + i√11) / 2 and x = (5 - i√11) / 2. Therefore, the solution set for the equation is {(5 + i√11) / 2, (5 - i√11) / 2}. This means that these two complex numbers are the only values of 'x' that will satisfy the original equation. The fact that we obtained complex solutions tells us that the parabola represented by the equation y = x² - 5x + 9 does not intersect the x-axis. This is a crucial connection between the algebraic solutions and the graphical representation of a quadratic equation. So, what are the key takeaways from this exercise? First, simplifying the equation is always the first step towards finding the solutions. Second, factoring is a useful technique, but it's not always applicable. Third, the quadratic formula is a powerful tool that guarantees a solution for any quadratic equation, even those with complex roots. And finally, understanding the nature of the solutions (real or complex) provides valuable insights into the behavior of the quadratic equation and its graph. Solving quadratic equations is a fundamental skill in algebra, and mastering these techniques will open doors to more advanced mathematical concepts. Remember, practice makes perfect, so keep working on different types of quadratic equations to build your confidence and expertise. You've done great work today, guys! Keep exploring the fascinating world of mathematics, and you'll be amazed at what you can achieve.