Solving Quadratic Equations: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of quadratic equations, specifically how to solve them. We'll break down the equation $x^2 - 10x = -34$, and figure out the correct answer from the multiple-choice options. So, let's get started and unravel this math mystery together! This journey will cover everything from understanding the basic concepts to applying different methods to arrive at the solution. We'll explore the quadratic formula, completing the square, and even touch upon complex numbers, ensuring that you grasp every single step. This explanation aims to simplify complex mathematical concepts, making them accessible and understandable for everyone. Whether you're a student struggling with algebra or just a math enthusiast, this guide has something for you.

First things first, what exactly is a quadratic equation? Well, it's an equation that can be written in the form $ax^2 + bx + c = 0$, where a, b, and c are constants, and a is not equal to zero. This is the standard form, which is super important! The highest power of the variable (in this case, x) is 2, hence the name "quadratic" (think "quad," like four, but related to the second power). Quadratic equations can have two solutions, one solution, or no real solutions (they might have complex solutions). Understanding this is key before starting. Now, looking at our equation, $x^2 - 10x = -34$, it doesn't look like the standard form yet. But don't worry, we can easily change that! Our goal is to manipulate the equation to match this standard form, which is crucial for solving it using different methods. Understanding the standard form is like having a map when navigating a new city – it provides a clear roadmap. We will be using this roadmap to guide our approach to finding the solutions of the given quadratic equation.

Before we begin solving, it's also helpful to quickly recap some core math concepts, like the basics of algebraic manipulation and the laws of exponents. This is just to make sure we're all on the same page. Remember, in algebra, we're basically playing a game of balancing equations. Whatever we do to one side, we must do to the other. This ensures that the equation remains true. This is the golden rule! Also, we will need to know about the properties of equality, where if a=b, then a+c=b+c, and so on. Understanding this principle ensures that the process of solving any equation maintains its integrity. Finally, remember that we have to isolate the unknown. By the end of this article, you will be able to easily identify the different parts of a quadratic equation and use the appropriate formula to find its solutions. Whether you're preparing for an exam or just brushing up on your math skills, this guide will provide you with the tools you need to succeed. So, grab your pencils, and let's get solving!

Step-by-Step Solution

Alright, let's solve the equation $x^2 - 10x = -34$. The first thing we need to do is to rewrite this equation into the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. To do this, we need to move the constant term from the right side to the left side by adding 34 to both sides of the equation. This gives us: $x^2 - 10x + 34 = 0$. Now we can easily identify our coefficients: $a = 1$, $b = -10$, and $c = 34$. Understanding these coefficients is essential, as they will be plugged into the quadratic formula. Making sure we have the proper understanding of a, b, and c values is crucial because any errors here will impact the correctness of the final solution. This process is like preparing the ingredients before cooking – knowing what you have will determine the recipe you can make. The standard form sets the stage for solving the quadratic equation. Now, we are ready to apply the quadratic formula. The quadratic formula is a universal tool designed to solve all quadratic equations.

Next, we'll use the quadratic formula to find the solutions for x. The quadratic formula is: x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}. Make sure you memorize this formula; it's your best friend when dealing with quadratic equations! Now, let's plug in the values of a, b, and c that we found earlier: x = rac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(34)}}{2(1)}. Let's break this down step-by-step to avoid any confusion. First, we have $-(-10)$, which simplifies to 10. Then, $(-10)^2$ is 100. And finally, $4(1)(34)$ equals 136. Substituting these values into the formula will provide us with a clearer picture of the calculation at each stage. So, we'll have x = rac{10 \pm \sqrt{100 - 136}}{2}.

Let's keep going. We need to simplify the expression under the square root: $100 - 136 = -36$. So now our formula looks like this: x = rac{10 \pm \sqrt{-36}}{2}. Here, we see that we have a negative number under the square root, which means we're going to get complex solutions. Remember, the square root of -1 is represented by i, the imaginary unit. Let's simplify this further: $\sqrt{-36} = \sqrt{36} * \sqrt{-1} = 6i$. Thus, our equation becomes x = rac{10 \pm 6i}{2}. The appearance of the imaginary unit indicates that we are dealing with complex numbers.

Finally, we simplify the expression. We can divide both the real and imaginary parts by 2. Thus, x = rac{10}{2} \pm rac{6i}{2}, which simplifies to $x = 5 \pm 3i$. And there we have it! The solutions to the equation $x^2 - 10x = -34$ are $x = 5 + 3i$ and $x = 5 - 3i$. These are the complex solutions to the quadratic equation. So, the correct answer from the multiple-choice options is B. $x = 5 \pm 3i$. Remember, complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. Complex numbers are important in various fields of mathematics and engineering, providing solutions to equations that would otherwise be unsolvable within the realm of real numbers.

Conclusion

So, there you have it! We've solved the quadratic equation $x^2 - 10x = -34$ step by step. We transformed the equation into the standard form, correctly applied the quadratic formula, and found that the solutions are complex numbers. This comprehensive walkthrough should equip you with the skills to solve similar problems confidently. If you're still having trouble, don't worry! Practice makes perfect. Try solving more quadratic equations on your own, and don't hesitate to revisit these steps whenever you need a refresher. Always remember to double-check your calculations, especially when dealing with negative signs and square roots. Good job, guys! Keep practicing, and you'll become a pro at solving quadratic equations in no time. If you have any further questions, feel free to ask! Understanding the fundamentals of quadratic equations is a key aspect of any comprehensive math education. Mastering this skill not only helps with academic performance but also builds a solid foundation for more advanced math concepts. Remember, with consistent practice and a clear understanding of the principles, you can excel in solving quadratic equations. Keep up the great work! Always remember that math is a journey. Each equation you solve is a milestone on your path to mathematical mastery.