Solving (a-x)(x-b)=3ax-5ab-x^2 A Step-by-Step Guide

Hey guys! Ever stumbled upon a math problem that looks like it’s written in another language? You stare at it, scratch your head, and wonder where to even begin? Well, today we’re diving deep into one of those problems – the equation (a-x)(x-b) = 3ax - 5ab - x^2. Don't worry; we're going to break it down step-by-step, so by the end of this article, you'll feel like a math whiz.

Deciphering the Equation: What's the Buzz About (a-x)(x-b)=3ax-5ab-x^2?

Let's start by understanding what this equation is all about. At first glance, it might seem like a jumble of letters and symbols, but it’s a classic algebraic equation with a lot going on under the surface. This equation involves variables (x), constants (a and b), and a mix of multiplication, subtraction, and squaring. The goal? To unravel the mystery of 'x' and find its possible values in terms of a and b. This type of problem often pops up in algebra courses and can be a real head-scratcher if you don’t have the right approach. We will explore the intricacies of this equation, focusing on how to simplify it, rearrange terms, and ultimately solve for the unknown variable, x. So, grab your favorite beverage, settle in, and let's embark on this mathematical adventure together. We're about to turn this seemingly daunting equation into a piece of cake.

The Initial Expansion: Laying the Foundation

Alright, let’s get our hands dirty and start breaking down this equation. The first step in conquering this beast is to tackle the left side: (a-x)(x-b). We need to expand this expression, and that means using the good old distributive property (also known as the FOIL method – First, Outer, Inner, Last). So, we'll take each term in the first set of parentheses and multiply it by each term in the second set. Ready? Let’s do it!

Expanding (a-x)(x-b):

• a * x = ax
• a * -b = -ab
• -x * x = -x^2
• -x * -b = bx

Now, let's put it all together: ax - ab - x^2 + bx. Awesome! We've successfully expanded the left side of the equation. But we're not done yet; we've just laid the foundation. Now, let's bring the whole equation back into the picture. We have ax - ab - x^2 + bx = 3ax - 5ab - x^2. It looks a little less intimidating now, doesn't it? The goal here is to simplify the equation as much as possible, so we can eventually isolate x and find its value. Simplifying equations is like tidying up a messy room – once everything is in its place, it's much easier to see what you're working with. Remember, math is all about taking complex problems and breaking them down into smaller, manageable steps. So, let’s keep moving forward and see what simplifications we can make!

Simplifying the Equation: Taming the Algebraic Jungle

Now that we've expanded the left side, it’s time to simplify the entire equation. We've got ax - ab - x^2 + bx = 3ax - 5ab - x^2. The name of the game here is to collect like terms and see if we can cancel anything out. Think of it like sorting your laundry – we want to group all the socks together, all the shirts together, and so on. In math terms, that means grouping all the x^2 terms, all the ax terms, and all the constant terms (those with just a and b) together. Let's start by noticing that we have a -x^2 term on both sides of the equation. What happens if we add x^2 to both sides? Poof! They cancel each other out. This is a fantastic move because it simplifies our equation significantly. Now we're left with ax - ab + bx = 3ax - 5ab. See how much cleaner it looks already? Next up, let's deal with the ax terms. We have ax on the left and 3ax on the right. To get them on the same side, we can subtract ax from both sides. This gives us -ab + bx = 2ax - 5ab. We're making progress, guys! We're slowly but surely isolating the terms with x in them. Our next target is the constant terms – the ones with just a and b. We have -ab on the left and -5ab on the right. To get all the constant terms on the right side, we can add ab to both sides. This leaves us with bx = 2ax - 4ab. We’re getting closer to isolating x, and the equation is looking much more manageable. Remember, simplification is key in algebra. By carefully collecting like terms and canceling out where possible, we transform a complex equation into a simpler, more solvable one. Now, let's move on to the next step: isolating x and finding its value.

Isolating 'x': The Quest for the Unknown

We've reached a crucial stage in our mathematical journey. Our equation is now simplified to bx = 2ax - 4ab. The mission, should you choose to accept it, is to isolate x. This means getting x all by itself on one side of the equation. To do this, we need to get all the terms with x on one side and everything else on the other. Currently, we have bx on the left and 2ax on the right. Let's bring the 2ax over to the left side by subtracting it from both sides. This gives us bx - 2ax = -4ab. Now, we have all the terms containing x on the left side. The next step is to factor out x. This means pulling x out as a common factor from the terms bx and -2ax. When we do this, we get x(b - 2a) = -4ab. We're almost there! We've got x multiplied by something, and we want x alone. The final step is to divide both sides of the equation by the expression in the parentheses, which is (b - 2a). So, we divide both sides by (b - 2a), and voilà! We have x = -4ab / (b - 2a). We’ve done it! We've successfully isolated x and found its value in terms of a and b. But hold on a second… Before we declare victory, we need to think about one little detail. What if (b - 2a) is equal to zero? Division by zero is a big no-no in mathematics. It’s like trying to divide a pizza into zero slices – it just doesn’t make sense. So, we need to consider the case where b - 2a = 0, which means b = 2a. If b is equal to 2a, our solution for x is undefined. This is an important caveat to keep in mind. So, the final answer, with all the details, is x = -4ab / (b - 2a), provided that b is not equal to 2a. High five! We've conquered this equation and navigated our way to the solution. Isolating x is a fundamental skill in algebra, and you've just nailed it. Now, let's take a step back and reflect on our journey.

The Final Solution and its Nuances: Unveiling the Complete Picture

So, after all that hard work, we've arrived at the final solution: x = -4ab / (b - 2a), with the crucial condition that b ≠ 2a. It's like reaching the summit of a mountain – you take a moment to appreciate the view and reflect on the journey. Let's break down what this solution actually tells us. The value of x depends entirely on the values of a and b. It's a relationship, a connection between these three variables. If you plug in different values for a and b, you'll get different values for x. Math is like that – it's all about relationships and how things connect. But let's not forget the important caveat: b cannot be equal to 2a. This is a critical condition. If b were equal to 2a, we would be dividing by zero, which is a big no-no in the math world. It's like trying to fit an elephant into a teacup – it just doesn't work. So, this condition is essential for our solution to be valid. Understanding these nuances is what separates a good math student from a great one. It's not just about finding the answer; it's about understanding the context, the limitations, and the underlying principles. Now, let’s think about how this type of problem might show up in the real world. While this specific equation might not be directly applicable to everyday situations, the skills we've used to solve it – expanding, simplifying, isolating variables – are crucial in many fields, from engineering to economics. Solving equations is like being a detective – you're given clues (the equation), and you have to use your skills to find the hidden answer (the value of x). It’s a skill that will serve you well in many areas of life. So, congratulations! You've not only solved a challenging equation, but you've also honed your problem-solving skills. And that, my friends, is what math is all about.

Real-World Applications: Where Does This Math Fit In?

Okay, we've successfully cracked the equation (a-x)(x-b) = 3ax - 5ab - x^2 and found our solution for x. But you might be wondering,