Solving Quadratic Equations Step By Step Consider 8x² - 48x = -104

Hey guys! Ever felt like quadratic equations are just a jumble of numbers and symbols? Don't worry; you're not alone. Many students find them a bit intimidating at first. But trust me, once you understand the basics, they become much easier to handle. In this article, we're going to break down a specific quadratic equation and guide you step-by-step through the process of solving it. We'll focus on transforming the equation into a more manageable form where the coefficient of the $x^2$ term is 1. This is a crucial step in many quadratic equation solving techniques, so let's dive in!

Transforming the Equation: Setting the Stage for Success

Our starting point is the equation $8x^2 - 48x = -104$. The goal here is to rewrite this equation in the form $x^2 + \square x = \square$, where the boxes represent the new coefficients we need to find. The key to achieving this is to make the coefficient of the $x^2$ term equal to 1. Currently, it's 8, so we need to get rid of that. How do we do it? Simple! We divide the entire equation by 8. Remember, whatever you do to one side of the equation, you must do to the other to maintain the balance. This is a fundamental principle in algebra, and it's super important to keep it in mind.

When we divide each term by 8, we get:

$(8x^2)/8 - (48x)/8 = -104/8$

Simplifying this gives us:

$x^2 - 6x = -13$

And there you have it! We've successfully transformed the equation into the desired form. Notice how the coefficient of $x^2$ is now 1. This form is much easier to work with when we want to complete the square or use other methods to find the solutions for x. By making this initial adjustment, we've set ourselves up for success in the next steps of solving the quadratic equation. This transformation is not just about making the numbers smaller; it's about simplifying the structure of the equation so that we can apply standard solution techniques more effectively. So, remember this trick – it's a game-changer!

Diving Deeper: Why This Transformation Matters

Now, you might be wondering, "Why did we bother doing this in the first place?" That's a great question! Transforming the equation to have a leading coefficient of 1 opens up a world of possibilities when it comes to solving quadratic equations. One of the most common and powerful techniques is completing the square. This method relies on creating a perfect square trinomial on one side of the equation, which can then be factored easily. But completing the square is significantly easier when the coefficient of the $x^2$ term is 1. Think of it like this: trying to complete the square with a coefficient other than 1 is like trying to assemble a puzzle with some of the pieces bent out of shape. It's much harder, and you're more likely to make mistakes. Getting that coefficient to 1 straightens out the pieces, making the puzzle much easier to solve.

Another reason this transformation is so important is that it simplifies the process of using the quadratic formula. The quadratic formula is a universal solution for any quadratic equation, but it can be a bit cumbersome to use. When the equation is in the form $x^2 + bx = c$, the values you need to plug into the formula become much clearer and easier to manage. It's like having a recipe where the ingredients are already measured out – you can just focus on the cooking! So, by making this initial transformation, we're not just making the numbers smaller; we're also making the entire solving process more streamlined and efficient. This is a key strategy for mastering quadratic equations, and it's something you'll use again and again.

Completing the Square: A Step-by-Step Guide

Okay, now that we've got our equation in the form $x^2 - 6x = -13$, let's talk about completing the square. This is a super useful technique for solving quadratic equations, and it's not as scary as it sounds, I promise! The basic idea behind completing the square is to manipulate one side of the equation into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form $(x + a)^2$ or $(x - a)^2$. Think of it like this: we're trying to build a square out of the terms we have, and we need to figure out what's missing to make it a perfect square.

So, how do we do it? Here's the trick: take half of the coefficient of the x term (which is -6 in our case), square it, and add it to both sides of the equation. Half of -6 is -3, and (-3) squared is 9. So, we're going to add 9 to both sides. Why 9? Because adding 9 will allow us to factor the left side into a perfect square. It's like adding the missing piece to our square puzzle!

Let's do it:

$x^2 - 6x + 9 = -13 + 9$

Now, the left side can be factored into $(x - 3)^2$, and the right side simplifies to -4:

$(x - 3)^2 = -4$

See how we've transformed the equation? We've gone from a somewhat messy quadratic to a simple squared term equal to a number. This is a huge step forward, and it makes the rest of the solving process much easier. By completing the square, we've created a situation where we can isolate x by taking the square root of both sides. This is a powerful technique, and it's something you'll use a lot in algebra and beyond.

Solving for x: Unveiling the Solutions

Now that we have $(x - 3)^2 = -4$, we're in the home stretch! The next step is to take the square root of both sides of the equation. Remember, when you take the square root, you need to consider both the positive and negative roots. This is because both a positive and a negative number, when squared, will give you a positive result. It's a crucial detail that's easy to overlook, but it can make all the difference in getting the correct solutions.

So, taking the square root of both sides, we get:

$x - 3 = ±√(-4)$

Now, here's where things get a little interesting. We have the square root of a negative number, which means our solutions will be complex numbers. Complex numbers involve the imaginary unit "i", where $i = √(-1)$. So, we can rewrite $√(-4)$ as $2i$ (since $√(-4) = √(4 * -1) = √(4) * √(-1) = 2i$).

Our equation now looks like this:

$x - 3 = ±2i$

To isolate x, we simply add 3 to both sides:

$x = 3 ± 2i$

And there you have it! We've found our solutions for x. We have two complex solutions: $x = 3 + 2i$ and $x = 3 - 2i$. These solutions tell us that the parabola represented by the original quadratic equation does not intersect the x-axis. It's a fascinating result, and it highlights the power of algebra to reveal hidden truths about mathematical relationships. By carefully following the steps of completing the square and taking square roots, we've successfully navigated the world of complex numbers and found the solutions to our quadratic equation. This is a testament to the beauty and elegance of mathematics!

The Big Picture: Why This Matters in the Real World

Okay, so we've solved a quadratic equation with complex solutions. That's cool and all, but you might be wondering, "When am I ever going to use this in real life?" That's a fair question! While you might not be solving equations with imaginary numbers every day, the principles and techniques we've used here are incredibly valuable in a wide range of fields.

Quadratic equations pop up all over the place in physics, engineering, and even economics. They're used to model the trajectory of a projectile, the shape of a suspension bridge, and the growth of populations. Understanding how to solve quadratic equations is essential for anyone working in these fields. Think about it: engineers use quadratic equations to design structures that can withstand stress and strain, physicists use them to predict the motion of objects, and economists use them to analyze market trends. The ability to manipulate and solve these equations is a fundamental skill for problem-solving in the real world.

Even if you're not planning on becoming a scientist or engineer, the problem-solving skills you develop by working with quadratic equations are transferable to many other areas of life. Learning to break down a complex problem into smaller, more manageable steps, thinking logically, and persevering through challenges are all skills that will serve you well in any career or situation. Solving quadratic equations is not just about finding the right answer; it's about training your mind to think critically and creatively. So, the next time you're faced with a tough problem, remember the steps we took to solve this quadratic equation – you might be surprised at how helpful they are!

Key Takeaways and Final Thoughts

So, guys, we've covered a lot in this article! We started with a quadratic equation, transformed it to make it easier to work with, completed the square, and solved for x, even venturing into the realm of complex numbers. That's quite an accomplishment! The key takeaway here is that solving quadratic equations is a process, and each step builds upon the previous one. By understanding the underlying principles and techniques, you can tackle even the most challenging equations with confidence.

Remember, the first step is often the most important – in our case, transforming the equation to have a leading coefficient of 1. This simple step opens up a world of possibilities and makes techniques like completing the square much more manageable. Completing the square is a powerful tool, and it's something you'll use again and again in your mathematical journey. And don't forget about those complex numbers! They might seem a bit strange at first, but they're an essential part of the mathematical landscape, and they have real-world applications in fields like electrical engineering and quantum mechanics.

But perhaps the most important lesson is that mathematics is not just about memorizing formulas and procedures; it's about developing problem-solving skills and a way of thinking. By working through examples like this, you're not just learning how to solve quadratic equations; you're learning how to approach any problem with a logical and creative mindset. So, keep practicing, keep exploring, and keep pushing yourself – you've got this!

Alright, folks, let's wrap things up. We've journeyed through the intricacies of a quadratic equation, from its initial form to its complex solutions. Remember, the equation we tackled was $8x^2 - 48x = -104$. Our mission was to rewrite it in the form $x^2 + \square x = \square$, and we successfully navigated the steps to achieve this. This transformation is more than just a mathematical exercise; it's a gateway to unlocking a deeper understanding of quadratic equations and their solutions.

The ability to manipulate equations, complete the square, and work with complex numbers are all valuable skills that will serve you well in your mathematical pursuits. But beyond the specific techniques, remember the bigger picture: mathematics is about problem-solving, critical thinking, and perseverance. Embrace the challenges, learn from your mistakes, and never stop exploring the fascinating world of numbers and equations. You've got the tools, the knowledge, and the potential to master quadratic equations and so much more. Keep up the great work!