Solving X^2 = 6x - 58 A Step-by-Step Guide

Hey everyone! Today, we're diving into the fascinating world of quadratic equations. If you've ever wondered how to solve equations that involve variables raised to the power of 2, you're in the right place. Specifically, we're going to tackle the equation x^2 = 6x - 58. Don't worry if it looks intimidating at first; we'll break it down step-by-step so you can master the techniques involved. Understanding quadratic equations is crucial in various fields, from physics and engineering to economics and computer science. So, let's get started and unlock the secrets behind solving these equations!

Understanding Quadratic Equations

So, what exactly is a quadratic equation? In simple terms, it's an equation that can be written in the form ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable we want to solve for. The highest power of 'x' in a quadratic equation is 2, which is what gives it the name "quadratic." Think of it like this: the 'quad' part hints at the square (power of 2). Now, why are these equations so important? Well, they pop up in all sorts of real-world scenarios. Imagine calculating the trajectory of a ball thrown in the air, designing the curve of a bridge, or even predicting population growth. Quadratic equations are the tools we use to model and understand these phenomena.

In our specific case, we have the equation x^2 = 6x - 58. To work with it effectively, we first need to rearrange it into the standard form, ax^2 + bx + c = 0. This involves moving all the terms to one side of the equation, leaving zero on the other side. So, let's subtract 6x from both sides and add 58 to both sides. This gives us x^2 - 6x + 58 = 0. Now it looks much more like the standard quadratic equation we discussed earlier. Identifying the coefficients 'a', 'b', and 'c' is our next step. In this equation, 'a' is the coefficient of x^2, which is 1 (since x^2 is the same as 1x^2). The coefficient 'b' is the number in front of x, which is -6. And finally, 'c' is the constant term, which is 58. Got it? Great! Knowing these coefficients is essential because they're the key ingredients we'll use in the next part, where we explore different methods to solve quadratic equations. Stick with me, and you'll be solving these like a pro in no time!

Methods to Solve Quadratic Equations

Alright, now that we know what a quadratic equation is and how to put it in the standard form, let's dive into the exciting part: solving them! There are several methods we can use, each with its own strengths and when to use them. We'll focus on three main techniques: factoring, completing the square, and the quadratic formula. Each method provides a unique approach to finding the values of 'x' that satisfy the equation. Understanding these methods gives you a versatile toolkit for tackling any quadratic equation that comes your way. So, let's get started and explore each method in detail.

Factoring

The first method we'll explore is factoring. Factoring is like reverse multiplication; we're trying to find two expressions that, when multiplied together, give us our original quadratic equation. This method is particularly useful when the quadratic equation can be easily broken down into two binomials. A binomial is just a mathematical expression with two terms, like (x + 2) or (x - 3). Factoring is often the quickest way to solve a quadratic equation, but it doesn't always work. Some equations are just too complex to factor easily. But when it does work, it's a breeze! So, how do we factor a quadratic equation? Let's take a closer look.

To factor a quadratic equation in the form ax^2 + bx + c = 0, we need to find two numbers that multiply to give 'c' and add up to 'b'. Think of it as a puzzle: we're searching for the right pieces that fit together. Once we find those numbers, we can rewrite the quadratic equation as a product of two binomials. For example, if we have the equation x^2 + 5x + 6 = 0, we need two numbers that multiply to 6 and add up to 5. Those numbers are 2 and 3, so we can factor the equation as (x + 2)(x + 3) = 0. Now, here's the cool part: if the product of two factors is zero, then at least one of the factors must be zero. This is called the zero-product property. So, we set each factor equal to zero and solve for 'x'. In our example, we have x + 2 = 0 and x + 3 = 0, which gives us x = -2 and x = -3. These are the solutions to the quadratic equation! Remember, factoring is a powerful tool, but it's not always the best choice. If you can't easily find the factors, don't worry; we have other methods up our sleeves.

Completing the Square

Next up, we have a method called completing the square. This technique might sound a bit mysterious at first, but it's a reliable way to solve any quadratic equation, even those that are difficult to factor. Completing the square involves manipulating the equation to create a perfect square trinomial on one side. A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, like (x + 3)^2 or (x - 4)^2. The beauty of this method is that it allows us to rewrite the equation in a form that's easy to solve. So, how does it work? Let's break it down step-by-step.

The first step in completing the square is to make sure the coefficient of x^2 is 1. If it's not, we divide the entire equation by that coefficient. Then, we move the constant term ('c') to the right side of the equation. Now comes the crucial part: we take half of the coefficient of 'x' (which is 'b'), square it, and add it to both sides of the equation. This step is what creates the perfect square trinomial on the left side. Why does this work? Well, by adding (b/2)^2, we're essentially filling in the missing piece to make the left side a perfect square. Once we have the perfect square trinomial, we can factor it as the square of a binomial. For example, if we have x^2 + 6x + 9, we can factor it as (x + 3)^2. Now, we have an equation in the form (x + something)^2 = some number. To solve for 'x', we take the square root of both sides, remembering to consider both the positive and negative square roots. Finally, we isolate 'x' by subtracting the constant term from both sides. And there you have it: the solutions to the quadratic equation! Completing the square might seem a bit more involved than factoring, but it's a valuable technique to have in your arsenal, especially when factoring isn't straightforward.

The Quadratic Formula

Last but definitely not least, we have the quadratic formula. This is the ultimate tool for solving quadratic equations. Think of it as the Swiss Army knife of quadratic equations – it works every time, no matter how complex the equation is. The quadratic formula is a mathematical expression that directly gives you the solutions to any quadratic equation in the form ax^2 + bx + c = 0. It's a bit like magic: you plug in the coefficients 'a', 'b', and 'c', and the formula spits out the answers. So, what is this magical formula? Let's take a look.

The quadratic formula is given by x = [-b ± √(b^2 - 4ac)] / (2a). Yes, it looks a bit intimidating, but don't worry, it's easier to use than it seems. The '±' symbol means we have two solutions: one where we add the square root and one where we subtract it. The expression inside the square root, b^2 - 4ac, is called the discriminant. The discriminant tells us a lot about the nature of the solutions. If the discriminant is positive, we have two distinct real solutions. If it's zero, we have one real solution (a repeated root). And if it's negative, we have two complex solutions. Complex solutions involve imaginary numbers, which we won't delve into in this article, but it's good to know they exist. To use the quadratic formula, we simply identify the coefficients 'a', 'b', and 'c' from our quadratic equation, plug them into the formula, and simplify. The result will be the solutions for 'x'. The quadratic formula is a lifesaver when factoring is difficult or completing the square seems too tedious. It's a reliable and powerful tool that ensures you can solve any quadratic equation that comes your way. So, let's remember our equation x^2 - 6x + 58 = 0 and use this formula. Get ready to see the magic in action!

Solving x^2 = 6x - 58 Using the Quadratic Formula

Alright, now that we've armed ourselves with the knowledge of different methods to solve quadratic equations, let's get back to our original problem: x^2 = 6x - 58. We've already transformed this equation into the standard form: x^2 - 6x + 58 = 0. Now, it's time to put the quadratic formula to work. Remember, the quadratic formula is x = [-b ± √(b^2 - 4ac)] / (2a). We need to identify the coefficients 'a', 'b', and 'c' from our equation. As we discussed earlier, 'a' is the coefficient of x^2, which is 1. The coefficient 'b' is the number in front of x, which is -6. And 'c' is the constant term, which is 58. Got those values? Great! Now, let's plug them into the formula.

Plugging the values into the quadratic formula, we get: x = [-(-6) ± √((-6)^2 - 4 * 1 * 58)] / (2 * 1). Take a deep breath; we're just crunching numbers now! Let's simplify this step-by-step. First, -(-6) becomes 6. Next, (-6)^2 is 36. And 4 * 1 * 58 is 232. So, our equation becomes x = [6 ± √(36 - 232)] / 2. Now, let's simplify the expression inside the square root. 36 - 232 is -196. Uh oh, we have a negative number under the square root! This means we're dealing with complex solutions, as we mentioned earlier. But don't worry, we can still handle this. The square root of -196 is 14i, where 'i' is the imaginary unit (√-1). So, our equation becomes x = [6 ± 14i] / 2. Finally, we can simplify this by dividing both the real and imaginary parts by 2. This gives us x = 3 ± 7i. And there you have it! The solutions to the quadratic equation x^2 = 6x - 58 are x = 3 + 7i and x = 3 - 7i. These are complex solutions, which means they have both a real part (3) and an imaginary part (7i and -7i). Congratulations! You've successfully solved a quadratic equation with complex solutions using the quadratic formula. You're well on your way to mastering quadratic equations!

Conclusion

Wow, we've covered a lot in this article! We started by understanding what quadratic equations are and why they're important. Then, we explored three powerful methods for solving them: factoring, completing the square, and the quadratic formula. Finally, we tackled our specific equation, x^2 = 6x - 58, and discovered that it has complex solutions: x = 3 + 7i and x = 3 - 7i. You've learned how to identify coefficients, apply different solution techniques, and interpret the results. Solving quadratic equations is a valuable skill that opens doors to understanding many real-world phenomena. Whether you're calculating trajectories, designing structures, or analyzing data, the ability to work with quadratic equations will serve you well.

Remember, practice makes perfect! The more you work with quadratic equations, the more comfortable and confident you'll become. Try solving different equations using various methods. Explore the patterns and connections between the coefficients and the solutions. Don't be afraid to make mistakes; they're part of the learning process. And most importantly, have fun! Quadratic equations might seem challenging at first, but they're also fascinating and rewarding to solve. So, keep exploring, keep learning, and keep unlocking the power of mathematics!