Simplify Algebraic Expressions: A Beginner's Guide

Simplifying the Expression: Unveiling the Solution

Hey guys! Let's dive into simplifying the algebraic expression: $(3x - 5)(4 - 9x) + (2x + 1)(6x^2 + 5)$. This might look a bit intimidating at first, but trust me, it's just a matter of careful expansion and combining like terms. We'll break it down step by step, making it super easy to follow. This process involves multiplying out the brackets and then collecting all the similar terms together. The key is to be organized and methodical. Don't worry, with a little practice, you'll be simplifying expressions like a pro! So, let's get started, and I'll walk you through each step so that you can ace it. Remember, algebra is all about patterns, so once you get the hang of it, you'll see how everything clicks into place. This kind of problem is a great example of the distributive property in action, which is a fundamental concept in algebra. Understanding this will open up a whole new world of problem-solving. It is important to note that the order of operations (PEMDAS/BODMAS) is crucial here to ensure we get the correct answer. This means we do Parentheses/Brackets first, then Exponents/Orders, then Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). Let's get into it!

First, we will start with the first part of the expression, which is $(3x - 5)(4 - 9x)$. To expand this, we use the distributive property. We multiply each term in the first bracket by each term in the second bracket. So, $(3x * 4) + (3x * -9x) + (-5 * 4) + (-5 * -9x)$. This simplifies to $12x - 27x^2 - 20 + 45x$. Now, let's do the second part of the expression. We have $(2x + 1)(6x^2 + 5)$. Similarly, we multiply each term in the first bracket by each term in the second bracket. This gives us $(2x * 6x^2) + (2x * 5) + (1 * 6x^2) + (1 * 5)$, which simplifies to $12x^3 + 10x + 6x^2 + 5$. Finally, we combine everything together: $(12x - 27x^2 - 20 + 45x) + (12x^3 + 10x + 6x^2 + 5)$. Now, we combine like terms, which means we add or subtract the terms with the same power of x. For instance, we combine the x terms, the x² terms, the constant terms, and the x³ terms. The final result is our simplified expression! Let's proceed. Make sure you understand the distributive property, it is a building block for algebra.

Step-by-Step Simplification Process

Alright, let's get our hands dirty and simplify this expression step by step! We'll break down each part to make sure we don't miss anything. Remember, accuracy is key in algebra, so take your time and double-check your work. The aim here is to meticulously expand the brackets and then neatly gather the like terms. We'll start by tackling the first part of the expression and then move on to the second, before putting everything together. Each step is important, and we'll try to make it as clear as possible. This problem uses fundamental algebra principles, so mastering this technique can go a long way in your overall mathematical prowess. You'll find that once you understand the principles, solving similar problems becomes a breeze. The ability to simplify expressions like this is not only essential in algebra but is also a fundamental skill needed in various fields like physics, engineering, and computer science. So, let's get started and unravel the secrets of this expression! We will also use the order of operations to ensure that everything is solved correctly. Make sure to always follow the proper order of operations. Let's get to it!

First, we'll focus on expanding $(3x - 5)(4 - 9x)$. Using the distributive property, we get: $3x * 4 - 3x * 9x - 5 * 4 - 5 * -9x$, which simplifies to $12x - 27x^2 - 20 + 45x$. Let's simplify this by combining like terms: $12x + 45x - 27x^2 - 20$, which gives us $57x - 27x^2 - 20$. Moving on to the second part, $(2x + 1)(6x^2 + 5)$, we get $2x * 6x^2 + 2x * 5 + 1 * 6x^2 + 1 * 5$, which simplifies to $12x^3 + 10x + 6x^2 + 5$. Now, we combine everything: $57x - 27x^2 - 20 + 12x^3 + 10x + 6x^2 + 5$. Combining like terms, we get $12x^3 - 27x^2 + 6x^2 + 57x + 10x - 20 + 5$, which simplifies to $12x^3 - 21x^2 + 67x - 15$. Voila! We've simplified the expression. Remember, the key is patience and careful execution of the distributive property. The final step involves combining like terms to arrive at a simplified form of the original equation. This will make you better at math. Remember to double check your work. Check each term, making sure you have accounted for every term from the initial expression. If you follow these simple steps, you'll get the hang of it in no time!

Detailed Breakdown of Expansion and Simplification

Let's delve deeper into the expansion and simplification process. We'll go through each step in even more detail so you can fully grasp the concepts. Understanding the reasoning behind each action is crucial to mastering algebra. This detailed explanation will give you the confidence to tackle similar problems on your own. Expanding the brackets and combining like terms are the most important things here. This detailed explanation should help you feel confident. This is like a playbook for algebra. This will help you. Let's break down our equation further!

Starting with $(3x - 5)(4 - 9x)$, we'll multiply each term in the first bracket by each term in the second bracket. This means:

$3x * 4 = 12x$ $3x * -9x = -27x^2$ $-5 * 4 = -20$ $-5 * -9x = 45x$

So, $(3x - 5)(4 - 9x)$ expands to $12x - 27x^2 - 20 + 45x$. Now, let's combine the like terms. We have two x terms, $12x$ and $45x$. Adding these, we get $57x$. So our expression becomes $57x - 27x^2 - 20$. Moving on to $(2x + 1)(6x^2 + 5)$, we do the same thing:

$2x * 6x^2 = 12x^3$ $2x * 5 = 10x$ $1 * 6x^2 = 6x^2$ $1 * 5 = 5$

So, $(2x + 1)(6x^2 + 5)$ expands to $12x^3 + 10x + 6x^2 + 5$. Now, we combine everything. We have $57x - 27x^2 - 20$ from the first expansion and $12x^3 + 10x + 6x^2 + 5$ from the second. Putting it all together, we have $12x^3 - 27x^2 + 6x^2 + 57x + 10x - 20 + 5$. Combining like terms again, we get $12x^3 - 21x^2 + 67x - 15$. This is our final answer! Remember to practice, and you will see it will get easier with each problem. Each step is important, so take your time. And that's all there is to it, guys! You are now a pro at simplifying these kinds of expressions! Keep practicing, and you'll become a master of algebraic manipulation. Make sure you follow each step to ensure accuracy.

Conclusion: Mastering Algebraic Simplification

Congratulations! You've successfully simplified a complex algebraic expression. By following the steps, using the distributive property and combining like terms, you've unlocked the ability to tackle similar problems. This skill is foundational in mathematics, opening doors to more advanced concepts. Remember, the more you practice, the more comfortable and confident you will become. Always double-check your work and be meticulous. So, keep practicing, and you'll find yourself mastering algebraic simplification in no time! This skill is incredibly useful and the key to success in more complex mathematics.

In summary, simplifying the expression $(3x - 5)(4 - 9x) + (2x + 1)(6x^2 + 5)$ involved expanding each set of brackets using the distributive property, and then combining like terms. The final answer is $12x^3 - 21x^2 + 67x - 15$. The whole process is easy if you focus on the process. Keep practicing and soon you will be a master. It is important to remember the distributive property. Now that you have the steps, you can do it yourself! Don't be intimidated by complex-looking expressions; with practice and the right approach, you can conquer them all. The ability to simplify expressions is a fundamental skill in algebra and is essential for solving equations and understanding more complex mathematical concepts. Keep practicing, and you'll be amazed at how quickly you improve. This skill is also going to make things much easier. Well done, and keep up the great work!