Unraveling Quadratic Equations: A Step-by-Step Guide

Hey everyone, let's dive into the fascinating world of quadratic equations! Today, we're going to break down how to solve an equation like (x+2)² + (x-3)² = 17. We'll use the trusty discriminant, find those all-important roots (x₁ and x₂), and make sure everything clicks into place. This stuff might seem intimidating at first, but trust me, with a little patience and the right approach, you'll be acing these problems in no time. So, grab your pencils, and let's get started. We will explore how to find the roots of the equation (x+2)²+(x-3)²=17, utilizing the quadratic formula and the discriminant. This will involve expanding the equation, simplifying it into the standard quadratic form (ax² + bx + c = 0), calculating the discriminant, and finally, determining the values of x₁ and x₂. This process isn't just about getting an answer; it's about understanding the underlying principles of quadratic equations. We’ll be transforming the given expression, using the discriminant, and arriving at the solutions for x. The goal here isn't just to find the answers, but to really understand the 'why' behind each step. Let's make sure we truly grasp each aspect of solving these kinds of problems, including applying the quadratic formula. Remember, the journey is as important as the destination! We will look into the general approach, which involves converting the equation into the standard quadratic form, calculating the discriminant, and then finding the roots. This method provides a clear, systematic way to solve the equation. The equation (x+2)²+(x-3)²=17, which, at first glance, may seem a bit complex, can be simplified using basic algebra principles. Let’s unravel the steps involved in solving such equations. We will convert the equation into the standard quadratic form, calculate the discriminant to understand the nature of the roots, and finally, find the values of x₁ and x₂. This is a journey through algebra, revealing how each component plays a role in finding the solution. So, stick around, follow along, and let's decode the steps to effectively solve this quadratic equation.

Step-by-Step Solution

Alright, let’s get down to business and actually solve this equation. The first thing we need to do is expand those squares. Remember the formula (a+b)² = a² + 2ab + b². Let's apply that to our equation. First, let's expand (x+2)². That gives us x² + 4x + 4. Then, let's expand (x-3)². That gives us x² - 6x + 9. Now, we can rewrite the original equation using these expanded expressions. Guys, it's really not that bad, right? We're just breaking things down into smaller, more manageable pieces. The key is to take it one step at a time. This process is all about making the equation easier to work with. Remember, we are aiming to get the equation into the standard form ax² + bx + c = 0. Expanding and simplifying are our first steps towards that goal. It's like building a puzzle – we need to put the pieces together correctly to reveal the full picture. So far, we've only expanded the terms. Next, we are going to simplify the equation, this will involve combining like terms, so let's get to it! We have two x² terms, a 4x and a -6x, and constants 4 and 9. Combining the x² terms, we get 2x². Combining the x terms, 4x - 6x equals -2x. Finally, we have the constants. The terms are 4 + 9 = 13. We'll include the 17 from the other side of the equation. Putting it all together, we now have 2x² - 2x + 13 = 17. Now that we've expanded and combined, we want to bring everything to one side to get it into the standard form. We will do this by subtracting 17 from both sides, which will give us 2x² - 2x - 4 = 0. This is starting to look much better, isn't it? Our equation is now in the standard form ax² + bx + c = 0. It's a huge step toward solving the quadratic equation. So, as we transform the equation to the standard form ax² + bx + c = 0, we're preparing it for the next steps. Next up, we will calculate the discriminant.

Calculating the Discriminant (D = b² - 4ac)

Now that we have the equation in standard form, it's time to find the discriminant. This is where the magic really starts to happen, guys! The discriminant helps us determine the nature of the roots. Remember, the formula is D = b² - 4ac. To find D, we'll use the coefficients from our standard form equation, 2x² - 2x - 4 = 0. In this equation, a = 2, b = -2, and c = -4. Now, let's plug these values into the discriminant formula. We have D = (-2)² - 4 * 2 * -4. Doing the math, we get D = 4 + 32, which simplifies to D = 36. That's great news, right? A positive discriminant tells us we're going to have two distinct real roots. So, the discriminant helps us understand what kind of roots we're dealing with. If the discriminant is positive, like in our case, we know there are two real and distinct solutions. If it were zero, we'd have one real solution (a repeated root). A negative discriminant, well, that's when things get into the complex number zone – no real solutions, just imaginary ones. So, the discriminant is our compass, guiding us on the nature of our solutions. With a value of D = 36, we know the equation has two unique solutions. The discriminant is crucial for understanding the character of our answers. The steps we have taken so far, have set us up for the final push: finding the values of x. Let's move on to using the quadratic formula to solve for those values.

Finding the Roots (x₁ and x₂)

Alright, time to find those roots! We'll use the quadratic formula: x = (-b ± √(D)) / 2a. We already know a = 2, b = -2, and D = 36. Let's plug these values into the quadratic formula. We get x = (2 ± √(36)) / (2 * 2). That simplifies to x = (2 ± 6) / 4. This is where we split it into two equations to find our two roots, x₁ and x₂. For x₁, we have (2 + 6) / 4, which is 8 / 4 = 2. For x₂, we have (2 - 6) / 4, which is -4 / 4 = -1. So, the roots of our equation are x₁ = 2 and x₂ = -1. Awesome, right? We have successfully solved the quadratic equation! The quadratic formula is a powerful tool. It gives us a direct way to find the roots, once we have a, b, and c, and the discriminant. x₁ = 2 and x₂ = -1 are the solutions that make the original equation true. These roots represent the points where the quadratic equation crosses the x-axis when graphed. These are the values of x that satisfy our original equation, and they are crucial for solving real-world problems. Understanding how to find x₁ and x₂ is a fundamental skill in algebra. Remember, the journey to finding x₁ and x₂ involves applying the quadratic formula, and breaking down the equation into simpler components. This also gives us a clear understanding of the relationships between the coefficients and the roots. Finally, we arrive at our solutions. We have successfully found the roots of the equation, which completes our task. Now, we've solved the equation and have the values of x₁ and x₂. Let’s wrap it up and summarize what we've learned.

Conclusion

Alright, guys, we made it! We successfully solved the quadratic equation (x+2)² + (x-3)² = 17. We started by expanding the equation, simplifying it to 2x² - 2x - 4 = 0. Then, we calculated the discriminant, finding D = 36, which told us we'd have two real roots. After that, we used the quadratic formula to find the roots x₁ = 2 and x₂ = -1. This example has given us a chance to understand the core steps in solving quadratic equations. This step-by-step approach not only helps in finding the solution but also deepens our comprehension. Remember the key steps: expand, simplify, find the discriminant, and apply the quadratic formula. With a bit of practice, you’ll be solving these equations with confidence. Mastering these steps is important because it is like learning a new language. You begin with the basics, and from there, you start stringing the words together to build sentences. Similarly, in solving quadratic equations, we begin with the basics and use each step to create a solution. The next time you encounter a quadratic equation, remember the steps we've taken today. You can break it down, apply the formulas, and find the solution. Each step plays a crucial role in providing clarity. You have the tools, so go out there and conquer those equations! Keep practicing, keep learning, and before you know it, you'll be a pro. The process of breaking down a problem into manageable steps is critical. This approach can be applied not just to math, but also to many aspects of life. Great work, everyone! Keep practicing, and you will get better. Thanks for joining me today, and I hope this helps you become a quadratic equation wizard! Remember, the goal isn't just to get the answer, but to understand the logic. This will surely assist you in your future endeavors. Always remember that learning takes time and patience, but with consistent efforts, you can master any skill.