Solving Quadratic Equations Finding Roots Of X²/2 = 5x - 17
Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of quadratic equations. We're going to tackle a specific equation: $ rac{x^2}{2}=5 x-17$. Our mission? To find its roots – those elusive values of 'x' that make the equation true. Finding roots is a fundamental concept in algebra, and mastering it opens doors to solving a wide range of problems in various fields, from physics and engineering to economics and computer science. So, buckle up, grab your thinking caps, and let's embark on this mathematical adventure together!
Rewriting the Equation: Setting the Stage for Success
Before we can even think about finding the roots, we need to get our equation into a more manageable form. The presence of the fraction and the terms scattered on both sides of the equation can make things look a bit intimidating. But don't worry, guys! We'll use our algebraic superpowers to transform it into the standard quadratic form, which is . This form is our best friend when it comes to solving quadratic equations. It's like having a universal key that unlocks the solution using methods like factoring, completing the square, or the quadratic formula. Our initial equation, $ rac{x^2}{2}=5 x-17$, looks a bit messy, right? Our first step is to eliminate that pesky fraction. To do this, we'll multiply both sides of the equation by 2. Remember, whatever we do to one side of the equation, we must do to the other to maintain the balance. This gives us: $x^2 = 10x - 34$. Much better! Now, we need to get all the terms on one side to achieve the standard quadratic form. We'll subtract 10x and add 34 to both sides, resulting in: $x^2 - 10x + 34 = 0$. Boom! We've successfully transformed our equation into the standard form. Now, we can clearly identify our coefficients: a = 1, b = -10, and c = 34. These values will be crucial when we use the quadratic formula later on. This rewriting process is a critical first step in solving any quadratic equation. It's like laying the foundation for a building – a solid foundation ensures a stable structure. By transforming the equation into the standard form, we've set ourselves up for success in the next stages of finding the roots. Remember, guys, a little bit of algebraic manipulation can go a long way in simplifying complex problems!
Choosing Our Weapon: Quadratic Formula to the Rescue
Now that we have our equation in the standard form (), it's time to choose our method for finding the roots. We have a few options at our disposal, including factoring and completing the square. However, in this case, factoring might be a bit tricky, and completing the square can be a bit cumbersome. So, we're going to call in the big guns: the quadratic formula! The quadratic formula is a powerful tool that can solve any quadratic equation, regardless of its complexity. It's like having a Swiss Army knife for quadratic equations – it always gets the job done. The formula itself looks a bit intimidating at first, but don't let it scare you. It's simply a recipe that we can follow step-by-step to find the roots. Here it is: $x = \frac-b \pm \sqrt{b^2 - 4ac}}{2a}$. See? Not so scary after all! Now, let's break down what each part of the formula means. The 'a', 'b', and 'c' are the coefficients we identified earlier from our standard form equation. Remember, a = 1, b = -10, and c = 34. The '±' symbol means that we'll have two possible solutions$, is particularly important. The expression inside the square root, , is called the discriminant. The discriminant tells us about the nature of the roots. If the discriminant is positive, we have two distinct real roots. If it's zero, we have one real root (a repeated root). And if it's negative, we have two complex roots. In our case, we'll see what happens when we plug in our values. The quadratic formula is our trusty weapon for this battle against the unknown. It provides a systematic and reliable way to find the roots, even when other methods fall short. So, let's grab our formula, plug in our coefficients, and see what secrets it reveals!
Plugging and Chugging: Applying the Quadratic Formula
Alright, guys, it's time to get our hands dirty and put the quadratic formula to work! We've got our equation, , and we've identified our coefficients: a = 1, b = -10, and c = 34. We also have our trusty quadratic formula: $x = \frac-b \pm \sqrt{b^2 - 4ac}}{2a}$. Now, it's just a matter of plugging in the values and simplifying. Think of it like following a recipe – we just need to carefully substitute the ingredients and follow the steps. Let's start by plugging in the values into the formula}2(1)}$. Notice how we've replaced 'a', 'b', and 'c' with their corresponding values. It's crucial to be careful with the signs, especially when dealing with negative numbers. Now, let's simplify step-by-step. First, we simplify the terms inside the square root}2}$. Next, we simplify the expression under the square root}2}$. Aha! We've encountered something interesting here. We have a negative number under the square root. This means that our roots will be complex numbers. Remember, the square root of a negative number is an imaginary number. The imaginary unit, denoted by 'i', is defined as . So, we can rewrite $\sqrt{-36}$ as $\sqrt{36} * \sqrt{-1} = 6i$. Now, let's substitute this back into our equation{2}$. Finally, we can simplify by dividing both the real and imaginary parts by 2: $x = 5 \pm 3i$. And there we have it! We've found the roots of our quadratic equation. They are complex numbers: and . This process of plugging and chugging might seem tedious, but it's a fundamental skill in algebra. The key is to be meticulous, pay attention to the signs, and simplify step-by-step. With practice, you'll become a pro at wielding the quadratic formula!
Unveiling the Complex Roots: A Deeper Dive
So, guys, we've successfully navigated the quadratic formula and arrived at our roots: and . But these aren't your everyday, run-of-the-mill numbers. These are complex numbers! Complex numbers are a fascinating extension of the real number system, and they pop up in various areas of mathematics, physics, and engineering. They might seem a bit mysterious at first, but let's break them down and understand what they represent. A complex number is generally expressed in the form , where 'a' is the real part and 'b' is the imaginary part, and 'i' is the imaginary unit (). In our case, the roots and both have a real part of 5 and imaginary parts of 3 and -3, respectively. Notice that the two roots are complex conjugates of each other. This is a common occurrence when quadratic equations have complex roots. Complex conjugates have the same real part but opposite imaginary parts. The fact that our roots are complex tells us something important about the graph of the quadratic equation. Remember, the roots of an equation are the points where the graph intersects the x-axis. Since our roots are complex, it means that the graph of the parabola does not intersect the x-axis. It hovers either entirely above or entirely below the x-axis. This is a visual representation of why we get complex roots – there are no real number solutions that make the equation equal to zero. Complex numbers might seem abstract, but they have very real-world applications. They're used in electrical engineering to analyze alternating current circuits, in quantum mechanics to describe the behavior of particles, and in signal processing to analyze and manipulate signals. Understanding complex roots is a crucial step in developing a deeper understanding of quadratic equations and their applications. It expands our mathematical toolkit and allows us to tackle a wider range of problems. So, embrace the complexity, guys! It's where the real mathematical magic happens!
Wrapping Up: Mastering the Art of Root Finding
Well, guys, we've reached the end of our journey to find the roots of the quadratic equation $ rac{x^2}{2}=5 x-17$. We started by rewriting the equation in the standard quadratic form, then we unleashed the power of the quadratic formula, and finally, we unveiled the complex roots. It's been quite the mathematical adventure! Let's recap the key steps we took: 1. Rewriting the equation: We transformed the given equation into the standard form, , which is essential for applying the quadratic formula. 2. Choosing the method: We opted for the quadratic formula because it's a reliable method for solving any quadratic equation. 3. Applying the quadratic formula: We carefully plugged in the coefficients and simplified the expression, paying close attention to the signs and order of operations. 4. Interpreting the roots: We discovered that the roots were complex numbers, which means the graph of the equation does not intersect the x-axis. This journey highlights the importance of a systematic approach to problem-solving in mathematics. By breaking down a complex problem into smaller, manageable steps, we can conquer even the most challenging equations. The quadratic formula is a powerful tool, but it's just one piece of the puzzle. Understanding the underlying concepts, such as the standard form of a quadratic equation and the nature of complex numbers, is crucial for truly mastering the art of root finding. Remember, guys, practice makes perfect! The more you work with quadratic equations and the quadratic formula, the more comfortable and confident you'll become. So, keep exploring, keep experimenting, and keep pushing your mathematical boundaries. The world of mathematics is vast and fascinating, and there's always something new to discover. Now that you've mastered this quadratic equation, you're well-equipped to tackle even more complex mathematical challenges. Go forth and conquer, my fellow math adventurers!