Solving The Math Problem: Detailed Explanation

Hey math enthusiasts! Let's dive into solving the equation: (-9x – 3) (8x - 9) + (-9x − 3) (-4x — 7). Don't worry, guys, it might look a bit intimidating at first, but we'll break it down step by step to make sure you understand every bit of it. We are going to explore this math problem in detail, making it super easy to follow along. So grab your pens and paper, and let's get started on this exciting mathematical journey!

This kind of problem falls under the umbrella of algebraic expressions, where we're dealing with variables (like 'x') and constants. Our goal is to simplify the expression, combining like terms and getting to a cleaner, more manageable form. Think of it like organizing a messy room – we're grouping similar items together to make everything neater and easier to understand. The key here is to apply the distributive property, which states that a(b + c) = ab + ac. This is our main tool for expanding the expressions and then, we'll combine the terms.

First, we'll address each part of the equation separately, using the distributive property. This means we multiply each term inside the first set of parentheses by each term in the second set of parentheses. This might sound like a lot of work, but trust me, with each step, the solution gets clearer. The main thing is to keep track of each step and not to get ahead of ourselves. Remember, the math is always straightforward. We just need to take it one step at a time. The distributive property is like a key that unlocks the equation, making it possible to simplify. In the end, we should be able to reach a definite conclusion and be happy with our work. So let’s not delay any further. We’ll first focus on the first part of the expression: (-9x – 3) (8x - 9). Then, we’ll move on to the second one: (-9x − 3) (-4x — 7).

Step-by-Step Breakdown: Conquering the Equation

Okay, let's get to the nitty-gritty of solving this problem. We'll methodically work through each component, ensuring you grasp every single step. This hands-on approach will equip you with the skills to tackle similar equations with confidence. It's like assembling a puzzle; once you have all the pieces, putting it together becomes a breeze. So, are you ready to solve it?

Step 1: Expanding the First Part

Let’s start with the first part of the equation: (-9x – 3) (8x - 9). We apply the distributive property here. This means we'll multiply -9x by both 8x and -9, and then multiply -3 by both 8x and -9. It's like distributing the love (or in this case, the multiplication) to each term inside the parentheses. So, (-9x * 8x) + (-9x * -9) + (-3 * 8x) + (-3 * -9). Simplifying, this gives us: -72x² + 81x - 24x + 27. See? Not too bad, right? We're systematically breaking it down, and it's starting to look a lot more manageable.

• (-9x * 8x): This results in -72x², because when you multiply x by x, you get x². Remember the rules of exponents, guys!* (-9x * -9): This results in +81x. A negative times a negative is a positive.* (-3 * 8x): This results in -24x.* (-3 * -9): This results in +27.

Now, let's proceed to the second part of the equation. We are making good progress, and we can start to see how everything is turning out, bit by bit. Just remember to maintain focus. Let’s keep going!

Step 2: Expanding the Second Part

Now, let's move on to the second part of the original equation: (-9x − 3) (-4x — 7). We again apply the distributive property. Multiply -9x by both -4x and -7, and then multiply -3 by both -4x and -7. This expands to: (-9x * -4x) + (-9x * -7) + (-3 * -4x) + (-3 * -7). Simplifying this gives us: 36x² + 63x + 12x + 21. See? We're on a roll! Just a few more steps and we'll have our final answer.

• (-9x * -4x): This gives us 36x².* (-9x * -7): This gives us +63x.* (-3 * -4x): This gives us +12x.* (-3 * -7): This gives us +21.

We're making steady progress here, guys. Next, we will combine the two expanded parts and combine like terms. The most important thing here is to make sure we don’t lose any value or number. Let’s do it!

Step 3: Combining and Simplifying

Now that we've expanded both parts of the expression, it's time to put everything together and simplify. We had: -72x² + 81x - 24x + 27 from the first part and 36x² + 63x + 12x + 21 from the second part. Combine them, the equation becomes: -72x² + 81x - 24x + 27 + 36x² + 63x + 12x + 21. Now, we group together the like terms (the x² terms, the x terms, and the constants). This is where things really start to simplify. Combining the x² terms (-72x² + 36x²), we get -36x². Combining the x terms (81x - 24x + 63x + 12x), we get 132x. And finally, combining the constants (27 + 21), we get 48. So, our simplified expression is: -36x² + 132x + 48.

• Combine x² terms: -72x² + 36x² = -36x²* Combine x terms: 81x - 24x + 63x + 12x = 132x* Combine constants: 27 + 21 = 48

We have successfully simplified the equation, guys! Let's summarize it to make sure we don't miss anything.

The Final Answer: Unveiling the Solution

Alright, after meticulously expanding, combining, and simplifying, we've arrived at our final answer. The simplified form of the original expression (-9x – 3) (8x - 9) + (-9x − 3) (-4x — 7) is -36x² + 132x + 48. Awesome, right? It's like we started with a complicated puzzle and now we have a clear, easy-to-understand solution. This simplified form is equivalent to the original expression, but much easier to work with, whether you're trying to solve for 'x' in a larger equation or just understanding the behavior of the expression.

Remember, the key to solving these kinds of problems is to be patient, methodical, and pay close attention to the details. Always double-check your work, especially when dealing with negative signs and multiplication. The more you practice, the easier it will become. And, hey, congratulations on sticking with it to the end. You've successfully navigated the math problem, and you should be proud of your accomplishment!

Key Takeaways: Mastering Algebraic Expressions

Let’s recap what we've learned and highlight some crucial points. This knowledge will serve you well in future math problems. Here's a quick rundown of the key concepts and techniques we used:

• The Distributive Property: This is your best friend when expanding expressions. Always remember to multiply each term inside the parentheses by the term outside.
• Combining Like Terms: Grouping the same variables (like x², x, and constants) together makes the expression simpler and easier to manage.
• Order of Operations: Don't forget to follow the order of operations (PEMDAS/BODMAS) when performing calculations. This ensures you do the operations in the correct sequence.
• Attention to Detail: Pay close attention to signs (positive and negative) and coefficients (the numbers in front of the variables). One small mistake can change the entire answer.
• Practice, Practice, Practice: The more problems you solve, the more comfortable you'll become with these concepts. Practice regularly to hone your skills.

Conclusion: Your Path to Math Mastery

So, there you have it, guys! We've successfully solved the equation and learned a lot along the way. Remember, math is like any other skill – the more you practice, the better you become. Don't be afraid to tackle challenging problems; each one is an opportunity to learn and grow. Keep practicing, and you'll become a math whiz in no time. If you got any questions, feel free to ask. Keep learning and keep exploring the amazing world of mathematics! Until next time, keep crunching those numbers and expanding your mathematical horizons. Cheers! I hope you liked it.