Solving X² = 16x - 65 Finding Solutions In Complex Form
Hey everyone! Today, we're diving deep into the fascinating world of quadratic equations. Specifically, we're going to tackle the equation x² = 16x - 65 and break down the steps to find its solutions. Quadratic equations might seem intimidating at first, but trust me, with a little guidance, you'll be solving them like a pro in no time. Think of this as your friendly guide to conquering quadratic equations!
Understanding Quadratic Equations
First things first, let's understand what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this case, 'x') is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants. Now, it is important to identify that our equation x² = 16x - 65 doesn't quite look like the standard form, does it? That's our first little challenge! To get it into the familiar ax² + bx + c = 0 format, we need to rearrange the terms. We'll do this by subtracting 16x and adding 65 to both sides of the equation. This gives us: x² - 16x + 65 = 0. Awesome! Now we're talking the language of quadratic equations. This form is crucial because it allows us to easily identify the coefficients a, b, and c, which we'll need for solving the equation. In our case, a = 1 (the coefficient of x²), b = -16 (the coefficient of x), and c = 65 (the constant term). See? We're already making progress! Quadratic equations are powerful tools used in various fields, from physics and engineering to economics and computer science. They help us model parabolic paths, optimize designs, and understand growth patterns. So, mastering these equations opens doors to solving real-world problems. There are a few ways to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Factoring is a great method when the equation can be easily broken down into two binomials. Completing the square is a technique that transforms the equation into a perfect square trinomial, making it easier to solve. And then there's the quadratic formula, which is a universal tool that works for all quadratic equations, regardless of their complexity. We'll be using the quadratic formula in this case, as it's a reliable method for finding the solutions, especially when dealing with complex roots.
Solving x² = 16x - 65
So, how do we solve x² = 16x - 65? As we established, we first need to rewrite the equation in standard form: x² - 16x + 65 = 0. Now, we're ready to roll! The key to solving this equation lies in the quadratic formula. This formula is your best friend when dealing with quadratic equations, especially those that don't factor easily. The quadratic formula states that for an equation in the form ax² + bx + c = 0, the solutions for x are given by: x = (-b ± √(b² - 4ac)) / 2a. Remember this formula, guys! It's a game-changer. It might look a bit intimidating with all those symbols, but trust me, it's just a matter of plugging in the values and doing the math. Now, let's identify our a, b, and c values again. In our equation x² - 16x + 65 = 0: a = 1, b = -16, and c = 65. We've got all the ingredients we need! Next, we carefully substitute these values into the quadratic formula. This is where paying attention to detail is crucial. A small mistake in substituting can lead to a wrong answer. So, let's take it slow and steady. Substituting, we get: x = (-(-16) ± √((-16)² - 4 * 1 * 65)) / (2 * 1). Notice how we're being careful with the signs, especially the negative signs. This is a common area for errors, so double-check your substitutions! Now, it's time to simplify. Let's start by simplifying the expression under the square root. We have (-16)² - 4 * 1 * 65, which equals 256 - 260, which simplifies to -4. Uh oh! We've got a negative number under the square root. What does this mean? It means our solutions are going to be complex numbers, involving the imaginary unit 'i', where i = √(-1). Don't worry, complex numbers are just another type of number, and we'll handle them with confidence! So, our equation now looks like this: x = (16 ± √(-4)) / 2. We can rewrite √(-4) as √(4 * -1), which is equal to 2i. This is where the imaginary unit 'i' comes into play. So, our equation further simplifies to: x = (16 ± 2i) / 2. Finally, we can divide both terms in the numerator by 2 to get our solutions: x = 8 ± i. And there you have it! We've found the solutions to the quadratic equation x² = 16x - 65. They are x = 8 + i and x = 8 - i. These are complex conjugate pairs, which is a common occurrence when solving quadratic equations with a negative discriminant (the part under the square root in the quadratic formula).
Identifying a and b in the Solutions x = a + bi and x = a - bi
Okay, so we've found the solutions: x = 8 + i and x = 8 - i. But the question asks us to express these solutions in the form x = a + bi and x = a - bi, and then identify the values of a and b. This is actually the easy part! We've already done the hard work of solving the equation. Now, we just need to match our solutions to the given form. Let's take a closer look at our solutions: x = 8 + i and x = 8 - i. Comparing these to the form x = a + bi, we can see that in the first solution, a corresponds to the real part, which is 8, and b corresponds to the imaginary part, which is 1 (since i is the same as 1i). Similarly, in the solution x = 8 - i, a is still 8, and b is -1. But the question specifically asks for the values of a and b in the general forms x = a + bi and x = a - bi. So, we focus on the magnitude of the imaginary part. Therefore, we can confidently say that a = 8 and b = 1. You see how the solutions neatly fit into the given forms? This is a common way to represent complex solutions to quadratic equations. The 'a' represents the real part of the complex number, and the 'b' represents the imaginary part. The ± sign indicates that we have a pair of complex conjugate solutions, which are numbers that have the same real part but opposite imaginary parts. Understanding this form helps us visualize complex numbers on the complex plane and perform operations with them. So, to summarize, we've successfully solved the quadratic equation x² = 16x - 65, found its complex solutions, and identified the values of a and b in the forms x = a + bi and x = a - bi. We've conquered another quadratic equation! Remember, practice makes perfect. The more you work with quadratic equations, the more comfortable you'll become with solving them. Keep practicing, and you'll be a quadratic equation master in no time!
Conclusion: Mastering Quadratic Equations
Great job, guys! You've made it to the end of our journey into solving quadratic equations. We've taken a seemingly complex problem, x² = 16x - 65, and broken it down into manageable steps. From understanding the standard form of a quadratic equation to applying the quadratic formula and identifying complex solutions, we've covered a lot of ground. Remember, the key to success in mathematics, and especially with quadratic equations, is practice. The more you work through problems, the more confident you'll become in your ability to solve them. Don't be afraid to make mistakes – they're a natural part of the learning process. Just learn from them, and keep going! Quadratic equations are fundamental tools in mathematics and have wide-ranging applications in various fields. By mastering these equations, you're not just learning a mathematical concept; you're developing problem-solving skills that will serve you well in many areas of life. So, keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is full of fascinating concepts and problems to solve. And with the right tools and mindset, you can conquer them all. Until next time, keep those quadratic equations in mind, and happy solving!
