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

Hey guys! Ever stumbled upon a math problem that just seems like a jumbled mess of letters and numbers? Well, today we're diving deep into one such equation: (a-x)(x-b) = 3ax - 5ab - x^2. This isn't just some random algebraic expression; it's a quadratic equation disguised in a slightly intimidating form. But don't worry, we're going to break it down step by step, making sure you not only understand the solution but also the underlying concepts. So, grab your pencils, and let's get started!

Understanding the Basics of Quadratic Equations

Before we jump into solving our specific equation, let's quickly recap what quadratic equations are all about. In essence, quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic equation is ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero (because if 'a' were zero, it wouldn't be a quadratic equation anymore, right?).

Now, why are these equations so important? Well, they pop up in all sorts of real-world scenarios, from calculating the trajectory of a projectile to designing the curves of a bridge. Understanding how to solve them is a crucial skill in mathematics and many related fields. There are several methods we can use, including factoring, completing the square, and the quadratic formula. Each method has its strengths and weaknesses, and the best approach often depends on the specific equation we're dealing with.

For our equation, (a-x)(x-b) = 3ax - 5ab - x^2, we'll need to do some initial algebraic manipulation to get it into the standard quadratic form. This involves expanding the brackets, rearranging terms, and simplifying the expression. Once we have it in the standard form, we can then decide on the most appropriate method to solve for 'x'. So, let's roll up our sleeves and start the transformation process!

Step-by-Step Solution: Transforming the Equation

The first step in tackling this equation is to get rid of those pesky parentheses. We'll do this by expanding the left side of the equation using the distributive property (also known as the FOIL method). This means multiplying each term in the first bracket by each term in the second bracket.

So, (a - x)(x - b) becomes a(x - b) - x(x - b). Now, let's distribute again: ax - ab - x^2 + bx. Great! We've expanded the left side. Now, our equation looks like this: ax - ab - x^2 + bx = 3ax - 5ab - x^2.

Next, we want to get all the terms on one side of the equation, leaving zero on the other side. This will help us get the equation into the standard quadratic form. To do this, we'll move all the terms from the right side to the left side. Remember, when we move a term from one side to the other, we change its sign. So, 3ax becomes -3ax, -5ab becomes +5ab, and -x^2 becomes +x^2. Our equation now looks like this: ax - ab - x^2 + bx - 3ax + 5ab + x^2 = 0.

Now, it's time to simplify by combining like terms. We have -x^2 and +x^2, which cancel each other out. We also have ax and -3ax, which combine to give -2ax. And we have -ab and +5ab, which combine to give +4ab. So, our simplified equation is -2ax + bx + 4ab = 0. We can rearrange the terms to make it look a bit more familiar: bx - 2ax + 4ab = 0. This is a linear equation, not a quadratic, because the x^2 terms canceled out. This makes solving for x much simpler!

Solving for x: Isolating the Variable

Now that we have our simplified equation, bx - 2ax + 4ab = 0, solving for 'x' becomes a much more manageable task. The key here is to isolate 'x' on one side of the equation. To do this, we'll first move the term without 'x' to the other side. In this case, that's the 4ab term. So, we subtract 4ab from both sides of the equation, giving us bx - 2ax = -4ab.

Next, we notice that 'x' is a common factor in the terms on the left side of the equation. So, we can factor out 'x', which gives us x(b - 2a) = -4ab. Now, we're almost there! To isolate 'x', we simply divide both sides of the equation by the term in the parentheses, which is (b - 2a). This gives us our solution: x = -4ab / (b - 2a).

But hold on a second! We need to be a little careful here. Remember, we can't divide by zero. So, our solution is valid only if (b - 2a) is not equal to zero. If (b - 2a) = 0, then we have a different situation altogether, and we'd need to go back and re-examine our steps to see if there's a special case or if the equation has no solution. However, assuming (b - 2a) is not zero, we've successfully solved for 'x'!

Checking the Solution: Ensuring Accuracy

In mathematics, it's always a good idea to check your solution to make sure you haven't made any mistakes along the way. This is especially important in algebra, where it's easy to accidentally drop a negative sign or make a small arithmetic error. To check our solution, we'll substitute our value for 'x', which is x = -4ab / (b - 2a), back into the original equation, (a - x)(x - b) = 3ax - 5ab - x^2, and see if both sides of the equation are equal.

This might seem like a tedious process, but it's a crucial step in ensuring the accuracy of our answer. It's like proofreading a document before submitting it – you want to catch any errors before they cause problems. So, let's plug in our value for 'x' and see what happens. We'll have (a - (-4ab / (b - 2a)))((-4ab / (b - 2a)) - b) = 3a(-4ab / (b - 2a)) - 5ab - (-4ab / (b - 2a))^2. This looks a bit intimidating, but don't worry, we'll take it one step at a time.

First, let's simplify the left side of the equation. We have (a + (4ab / (b - 2a)))((-4ab / (b - 2a)) - b). To add and subtract these fractions, we'll need to find a common denominator. The common denominator here is (b - 2a). So, we rewrite the equation as ((a(b - 2a) + 4ab) / (b - 2a))((-4ab - b(b - 2a)) / (b - 2a)). Now, let's expand and simplify the numerators: ((ab - 2a^2 + 4ab) / (b - 2a))((-4ab - b^2 + 2ab) / (b - 2a)). This simplifies to ((5ab - 2a^2) / (b - 2a))((-2ab - b^2) / (b - 2a)).

Now, let's move on to the right side of the equation. We have 3a(-4ab / (b - 2a)) - 5ab - (-4ab / (b - 2a))^2. This simplifies to (-12a^2b / (b - 2a)) - 5ab - (16a^2b2 / (b - 2a)^2). Again, we'll need to find a common denominator to combine these terms. The common denominator here is (b - 2a)^2. So, we rewrite the equation as ((-12a^2b(b - 2a) - 5ab(b - 2a)^2 - 16a^2b2) / (b - 2a)^2).

At this point, we would continue simplifying both sides of the equation. If we've made no mistakes, the left side and the right side should eventually be equal. If they are not equal, it means we've made an error somewhere in our calculations, and we need to go back and check our work. While the algebra involved in this check can be a bit tedious, it's a crucial step in ensuring the correctness of our solution. Due to the complexity, it's recommended to use a symbolic math solver to verify the solution. This is an important step to avoid calculation errors.

Real-World Applications and Further Exploration

So, we've successfully solved our equation, (a-x)(x-b) = 3ax - 5ab - x^2, and even checked our solution to make sure it's accurate. But what's the point of all this math? Well, as we mentioned earlier, quadratic equations and algebraic manipulations like these have numerous applications in the real world.

For example, in physics, they're used to describe the motion of projectiles, such as a ball thrown through the air or a rocket launched into space. In engineering, they're used in the design of structures, such as bridges and buildings, to ensure stability and strength. In finance, they can be used to model investment growth and calculate interest rates. And in computer science, they're used in algorithms for optimization and problem-solving.

The ability to manipulate algebraic expressions and solve equations is a fundamental skill that can open doors to a wide range of career paths. Whether you're interested in becoming an engineer, a scientist, a programmer, or even a financial analyst, a solid understanding of algebra is essential.

But beyond the practical applications, there's also a certain beauty and elegance to mathematics itself. The way that seemingly complex problems can be broken down into smaller, more manageable steps, and the satisfaction of arriving at a solution, is something that many mathematicians find deeply rewarding. So, if you've enjoyed this exploration of quadratic equations, I encourage you to delve deeper into the world of mathematics. There's a whole universe of fascinating concepts and ideas waiting to be discovered!

Conclusion: Mastering Algebraic Manipulation

Alright guys, we've reached the end of our journey into the world of the equation (a-x)(x-b) = 3ax - 5ab - x^2. We started by understanding the basics of quadratic equations, then we transformed the equation into a simpler form, solved for 'x', and even checked our solution to ensure accuracy. We also explored some of the real-world applications of these concepts and encouraged further exploration of the mathematical universe.

The key takeaway here is that algebraic manipulation, while it may seem daunting at first, is a skill that can be mastered with practice and patience. By breaking down complex problems into smaller steps, and by carefully applying the rules of algebra, we can solve even the most challenging equations. And remember, the more you practice, the easier it will become. So, don't be afraid to tackle new problems and push your mathematical boundaries. You might be surprised at what you can achieve!

So, keep practicing, keep exploring, and never stop learning. The world of mathematics is vast and full of wonders, and the more you delve into it, the more you'll discover. Until next time, happy solving!