Solving (2x-3)(3x-4)-(x-73)(x-4)=40 A Step-by-Step Guide

Hey guys! Ever feel like you're staring at an equation that looks like it's written in another language? We've all been there! Today, we're tackling a math problem that might seem intimidating at first glance, but trust me, we can totally crack it together. The equation we're going to solve is: (2x - 3)(3x - 4) - (x - 73)(x - 4) = 40. Sounds intense, right? But don't worry, we'll break it down step-by-step so it's super easy to understand. Math can be like a puzzle, and this time we're going to find all the pieces and put them together.

Understanding the Equation and Why It Matters

Before we jump into solving, let's take a moment to understand what this equation actually means. At its heart, it's an algebraic equation, meaning it involves variables (in this case, 'x') and constants, combined with mathematical operations. The goal is to find the value (or values) of 'x' that make the equation true. In simpler terms, we're looking for the secret number(s) that, when plugged in for 'x', will make both sides of the equation equal. But why is this important? Well, equations like this pop up everywhere in real-world applications. They help us model situations in physics, engineering, economics, and even computer science. So, mastering the skill of solving equations is like unlocking a powerful tool that you can use in countless ways. Think about it: designing bridges, predicting stock market trends, or even creating the next viral video game – all of these could involve solving equations similar to this one. This isn't just about numbers and symbols; it's about understanding the relationships between things and making predictions about the world around us. So, stick with me, and let's turn this intimidating equation into a solvable problem. We're not just memorizing steps here; we're building a foundation for understanding how the world works!

Step-by-Step Solution: Conquering the Equation

Okay, let's dive into the nitty-gritty of solving this equation. Remember, the key is to take it one step at a time. We'll start by expanding the products, then simplifying, and finally isolating 'x'. It's like building a house: you need a solid foundation before you can put up the walls and roof. First, let's tackle the left side of the equation: (2x - 3)(3x - 4) - (x - 73)(x - 4) = 40. Our first mission is to expand the expressions (2x - 3)(3x - 4) and (x - 73)(x - 4). Remember the FOIL method? (First, Outer, Inner, Last) This will be our best friend here. Let's start with (2x - 3)(3x - 4). Multiplying the First terms (2x * 3x) gives us 6x². The Outer terms (2x * -4) result in -8x. The Inner terms (-3 * 3x) give us -9x, and the Last terms (-3 * -4) produce +12. So, (2x - 3)(3x - 4) expands to 6x² - 8x - 9x + 12. Now, let's expand (x - 73)(x - 4). The First terms (x * x) give us x². The Outer terms (x * -4) result in -4x. The Inner terms (-73 * x) give us -73x, and the Last terms (-73 * -4) produce +292. Thus, (x - 73)(x - 4) expands to x² - 4x - 73x + 292. See? We're already making progress! We've broken down the complicated multiplication into smaller, manageable pieces. Now, let's put it all together and simplify. This is where the real magic happens!

Simplifying and Combining Like Terms

Now that we've expanded the expressions, the next crucial step is simplification. This involves combining like terms – terms that have the same variable and exponent. Think of it like sorting your laundry: you put all the shirts together, all the pants together, and so on. We're doing the same thing here, but with mathematical terms. Remember, our expanded equation looks like this: 6x² - 8x - 9x + 12 - (x² - 4x - 73x + 292) = 40. Before we start combining, let's distribute that negative sign in front of the second set of parentheses. This is a super important step, because forgetting to distribute the negative is a common mistake that can throw off your entire solution. Distributing the negative sign, we get: 6x² - 8x - 9x + 12 - x² + 4x + 73x - 292 = 40. Now, the fun part: combining like terms! Let's start with the x² terms. We have 6x² and -x². Combining these gives us 5x². Next, let's move on to the x terms. We have -8x, -9x, +4x, and +73x. Adding these together, we get (-8 - 9 + 4 + 73)x, which simplifies to 60x. Finally, let's combine the constant terms: +12 and -292. This gives us -280. So, after simplifying, our equation looks much cleaner: 5x² + 60x - 280 = 40. We're getting closer to our goal! We've taken a messy equation and transformed it into a simpler, more manageable form. This is the power of algebra – we can manipulate expressions to make them easier to work with. Now, let's move on to the next step: setting the equation to zero.

Setting the Equation to Zero and Solving for x

We're on the home stretch! Our equation is looking much simpler now: 5x² + 60x - 280 = 40. To solve for 'x', we need to get all the terms on one side of the equation and set it equal to zero. This is a standard technique for solving quadratic equations (equations with an x² term). Think of it like balancing a scale: we want to make sure both sides are equal. To get zero on the right side, we need to subtract 40 from both sides of the equation. This gives us: 5x² + 60x - 280 - 40 = 40 - 40, which simplifies to 5x² + 60x - 320 = 0. Great! Now we have a quadratic equation in the standard form: ax² + bx + c = 0, where a = 5, b = 60, and c = -320. There are several ways to solve quadratic equations. We could try factoring, using the quadratic formula, or even completing the square. Factoring is often the quickest method if the equation factors nicely. However, in this case, the numbers are a bit large, so let's make our lives easier by dividing the entire equation by 5. This is allowed because we're doing the same operation to both sides of the equation, maintaining the balance. Dividing by 5, we get: x² + 12x - 64 = 0. Now, this looks much more manageable! We're one step closer to finding those elusive values of 'x'. The next step is to choose our weapon of choice: factoring or the quadratic formula. Let's see if we can factor this equation. We need to find two numbers that multiply to -64 and add up to 12. This is like detective work – we're looking for the clues that will unlock the solution. Can you think of two numbers that fit the bill? Stay tuned, because we're about to reveal the answer and finally solve for 'x'!

Factoring the Quadratic Equation and Finding the Solutions

Alright, let's crack this quadratic equation! We've simplified it to x² + 12x - 64 = 0. As we discussed, we're looking for two numbers that multiply to -64 and add up to 12. After a little thought (or maybe some trial and error), you might realize that the numbers 16 and -4 fit the criteria perfectly. 16 multiplied by -4 equals -64, and 16 plus -4 equals 12. Eureka! Now we can factor the quadratic equation. Factoring is like reverse-distributing: we're writing the quadratic expression as a product of two binomials. Using the numbers we found, we can rewrite the equation as: (x + 16)(x - 4) = 0. This is a huge step! We've transformed the equation into a form where we can easily find the solutions. Here's the key concept: if the product of two factors is zero, then at least one of the factors must be zero. In other words, either (x + 16) = 0 or (x - 4) = 0. This gives us two simple equations to solve. Let's solve them one at a time. If x + 16 = 0, then subtracting 16 from both sides gives us x = -16. This is our first solution! If x - 4 = 0, then adding 4 to both sides gives us x = 4. This is our second solution! And there you have it! We've successfully solved the equation. The solutions are x = -16 and x = 4. This means that if you plug either -16 or 4 back into the original equation, it will hold true. We've conquered the equation, one step at a time. Remember, solving equations is like building a muscle: the more you practice, the stronger you get. So, keep tackling those math problems, and you'll be amazed at what you can achieve. Now, let's take a moment to recap the key steps we followed to solve this equation. This will help solidify your understanding and make you an equation-solving pro!

Reviewing the Steps and Building Confidence

Wow, we made it! We successfully solved that equation, and that's something to be proud of. Let's take a moment to recap the journey we took, from the initial daunting equation to the final triumphant solution. This will not only reinforce your understanding but also build your confidence for tackling future challenges. Remember, the key steps were:

1. Expanding the products: We used the FOIL method (First, Outer, Inner, Last) to expand the expressions (2x - 3)(3x - 4) and (x - 73)(x - 4). This involved multiplying each term in the first binomial by each term in the second binomial.
2. Simplifying and combining like terms: After expanding, we combined terms with the same variable and exponent. This made the equation much cleaner and easier to work with.
3. Setting the equation to zero: We subtracted 40 from both sides of the equation to get all the terms on one side and zero on the other. This is a crucial step for solving quadratic equations.
4. Factoring the quadratic equation: We looked for two numbers that multiply to the constant term (-64) and add up to the coefficient of the x term (12). Once we found those numbers (16 and -4), we could rewrite the quadratic expression as a product of two binomials: (x + 16)(x - 4) = 0.
5. Finding the solutions: We used the principle that if the product of two factors is zero, then at least one of the factors must be zero. This allowed us to set each factor equal to zero and solve for x, giving us the solutions x = -16 and x = 4.

See? When we break it down step-by-step, it's not so scary after all! The most important thing is to be patient with yourself, practice regularly, and don't be afraid to ask for help when you need it. Math is a journey, not a destination. Every equation you solve is a step forward, and with each step, you'll become more confident and capable. So, keep practicing, keep exploring, and keep challenging yourself. You've got this! And remember, if you ever get stuck, there are tons of resources available, including online tutorials, math forums, and, of course, your friendly neighborhood math enthusiasts (like me!).

Practice Makes Perfect: Next Steps and Resources

Congratulations on conquering that equation! But like any skill, solving equations takes practice. The more you do it, the more natural it will become, and the more confident you'll feel. So, what's next? Here are a few ideas for taking your equation-solving skills to the next level:

• Practice, practice, practice! The best way to improve is to work through lots of examples. Look for practice problems in your textbook, online, or even create your own. Start with simpler equations and gradually work your way up to more challenging ones.
• Explore different methods: We solved this equation by factoring, but there are other methods, such as the quadratic formula and completing the square. Learning these different techniques will give you more tools in your toolbox and help you tackle a wider range of problems.
• Seek out resources: There are tons of amazing resources available online, including websites like Khan Academy, Wolfram Alpha, and YouTube channels dedicated to math tutorials. Don't hesitate to use these resources to deepen your understanding and get help with tricky concepts.
• Join a study group: Working with others can be a fantastic way to learn and stay motivated. You can bounce ideas off each other, explain concepts, and help each other through tough problems.
• Don't be afraid to ask for help: If you're struggling with a particular concept or problem, don't hesitate to ask your teacher, a tutor, or a friend for help. There's no shame in admitting you need assistance, and getting help early can prevent frustration and build your confidence.

Remember, math is a skill that builds over time. Be patient with yourself, celebrate your successes, and don't get discouraged by setbacks. Every mistake is a learning opportunity, and every problem you solve makes you stronger. So, keep practicing, keep learning, and keep challenging yourself. You've already proven that you can tackle tough equations. Now go out there and conquer the world of math!