Solving Quadratic Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of quadratic equations. Today, we're going to tackle a problem that might seem a bit intimidating at first, but trust me, it's totally manageable. Our goal is to transform the given equation, making it easier to solve. This is super important because it lays the groundwork for understanding more complex math concepts down the road. So, buckle up, and let's get started! We'll break down the process step by step, so you can follow along easily. No prior knowledge is required. We're going to take this problem and make it super easy to grasp. Think of this as a helpful guide to solving quadratic equations, making sure you understand each part of the process. We will simplify the equation so that the coefficient 'a' becomes 1. This is done by dividing all the terms by the original coefficient, in this case, 8. So, as you can see, understanding the problem is the first step toward its solution. Remember, the more you practice, the better you'll get. It's like riding a bike; you might wobble at first, but soon, you'll be cruising along smoothly. Let's get our hands dirty and solve the equation! Are you ready to roll? I know you are. Let's begin.

Simplifying the Initial Equation

Okay, let's start with the equation: $8x^2 - 48x = -104$. The first thing we want to do is to get the coefficient of the $x^2$ term (which we denote as 'a') to be 1. Why? Because it simplifies things and makes the equation easier to work with. To do this, we'll divide every term in the equation by 8. This is totally allowed, as long as we do it to every single term, so we maintain the equation's balance. When we divide $8x^2$ by 8, we get $x^2$. Dividing $-48x$ by 8 gives us $-6x$. Finally, dividing $-104$ by 8, we get $-13$. So, our equation becomes $x^2 - 6x = -13$. This is the first major step, and now we're one step closer to making it look the way we want. Remember, we're aiming for an equation in the form of $x^2 + ext{something} imes x = ext{something else}$.

Now, the equation looks much simpler. Before we proceed, it's worth noting why this is a good approach. By making 'a' equal to 1, we're setting the stage for easier factoring or completing the square. This is the basis for solving quadratic equations, a crucial skill in algebra. If we get stuck, we can go back and review this part again. We can always pause and rewind if you need. The key is to be patient and persistent. The more you practice, the more comfortable you'll become with this. So, just keep at it! You'll be surprised how quickly you pick up the concepts. This step is crucial because it allows us to use various solving methods like factoring, completing the square, or the quadratic formula more effectively. Let's not forget to practice it as we go. The main thing is we understand what to do to solve it. It's easy if you understand how to solve it. If we have this step down, we can move forward to the next step and solve it to get the final result. This is how we build up our skills one step at a time. This also helps in getting the correct result. So, are you ready to move on?

Rewriting in the Desired Form

We're almost there! Let's take another look at the equation we have: $x^2 - 6x = -13$. We want to rewrite it in the form $x^2 + ext{something} imes x = ext{something else}$. In our current equation, the 'something' multiplied by x is -6. So, we can directly see that the coefficient of x is -6. Thus, the blank in the requested format $x^2 + ext{blank} imes x = ext{blank}$ corresponds to -6. This is a direct match. Now, let's rewrite the equation to match the requested form. So, for the left side of the equation, it should be $x^2 + (-6)x$. This is the same as $x^2 - 6x$. And the right side remains unchanged, which is -13. The equation that fits the format of $x^2 + ext{blank} imes x = ext{blank}$ is therefore: $x^2 + (-6)x = -13$. This matches our target format perfectly! Now, we can directly see the values that fill in the blanks. The first blank is -6, and the second blank is -13. So, we have successfully rewritten the equation as requested. We have now simplified the original equation and converted it into the desired format, and we're ready to fill in the blanks.

This step is important because it clearly shows the structure of a quadratic equation in a simplified form, making it easier to manipulate and solve. The goal is to have the leading coefficient of $x^2$ be 1, which allows for a more straightforward process when using methods like completing the square or applying the quadratic formula. By recognizing this form, we can see what the missing values are, directly and effectively.

Filling in the Blanks

We've done all the hard work, guys! Now, we just need to fill in those blanks. Remember, our transformed equation is $x^2 - 6x = -13$. When comparing this to the target form x^2 + oxed{ ext{ }}x = oxed{ ext{ }}, we can easily identify the values. In the original equation, the coefficient of x is -6. This goes into the first blank. And the constant term on the right side of the equation is -13, which goes into the second blank. So, the filled-in equation is $x^2 + (-6)x = -13$. Thus, the answer is -6 for the first blank and -13 for the second. We've successfully rewritten the equation in the desired format and identified the missing values. That's it! We've gone from a somewhat messy initial equation to one that's easy to understand and prepare for further solving.

This is just one small piece of the puzzle when you're dealing with quadratic equations. Being able to manipulate equations and put them into a standard format is a key skill. It's like learning the alphabet before you start writing a novel. Without these basic skills, more advanced topics would be almost impossible to grasp. Remember, practice makes perfect. Don't be discouraged if you don't get it right away. Keep trying, and you'll get there. The sense of accomplishment you feel when you solve a problem is unbeatable. That's the beauty of mathematics; it is challenging, and it's rewarding. So, keep up the good work!

Conclusion: The Rewritten Equation

Alright, folks, let's recap what we've done. We started with the equation $8x^2 - 48x = -104$. We wanted to rewrite it in the form where the coefficient of the $x^2$ term is 1. We did this by dividing the entire equation by 8. This gave us $x^2 - 6x = -13$. Comparing this to the desired format of x^2 + oxed{ ext{ }}x = oxed{ ext{ }}, we found that the equation can be rewritten as $x^2 + (-6)x = -13$. Therefore, the first blank is -6, and the second blank is -13. Congratulations! You have successfully rewritten the equation to match the requested format.

This process might seem simple, but it forms the core of more advanced techniques in solving quadratic equations. By understanding how to manipulate equations, you're building a solid foundation for future mathematical concepts. Keep practicing, keep learning, and don't be afraid to ask questions. Keep in mind that every step builds on the previous ones, and each problem you solve makes you stronger. Math is a journey. Sometimes you get stuck, and other times, things just click. The most important thing is to keep trying. That is the key to success in everything. So, keep up the amazing work. You got this!