Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey guys! Let's dive into a fun algebra problem. We're going to simplify and evaluate an algebraic expression. This is a common task in algebra, and it's super important to understand the steps involved. Don't worry, it's not as scary as it looks. We'll break it down into easy-to-follow steps. Ready to get started? We will solve the expression: 2(a²-5ab+4b²)-3(2a²-2ab+3b²) where a=1.1 and b=0.8. The goal is to first simplify the expression and then substitute the values of a and b to find the final numerical answer. This process helps build a strong foundation in algebra. It emphasizes the importance of following the order of operations and accurately combining like terms. You'll often encounter similar problems, so getting comfortable with this type of question will really help you with future, more complex algebraic challenges. The most crucial part of this is to keep track of the signs (+ and -). That's often where people make mistakes. We'll be super careful and methodical. We'll work through it together, step-by-step, making sure you understand each part.

First, let's look at the given expression: 2(a² - 5ab + 4b²) - 3(2a² - 2ab + 3b²). We need to simplify this. The first step is to distribute the numbers outside the parentheses to each term inside the parentheses. This is where we apply the distributive property. It's essentially multiplying each term inside the parentheses by the number outside. Let's do that for the first part of the expression: 2(a² - 5ab + 4b²). We multiply each term by 2: 2 * a² = 2a², 2 * -5ab = -10ab, and 2 * 4b² = 8b². So, the first part becomes 2a² - 10ab + 8b². Now, let's do the same for the second part: -3(2a² - 2ab + 3b²). Remember to multiply each term by -3: -3 * 2a² = -6a², -3 * -2ab = 6ab, and -3 * 3b² = -9b². The second part becomes -6a² + 6ab - 9b². Now, let's put it all together. Our expression has now become: 2a² - 10ab + 8b² - 6a² + 6ab - 9b². Wow, that's a lot, right? Don't worry, we are not done yet, now it's time to simplify the expression by combining like terms.

Combining Like Terms: The Heart of Simplification

Alright, now that we've distributed the numbers, the next step is to combine like terms. This means grouping together terms that have the same variables raised to the same powers. For example, a² and a² are like terms, as are ab and ab, and b² and b². We will do it together, don't worry. To do this systematically, let's first identify all the a² terms: 2a² and -6a². When we combine these, we get 2a² - 6a² = -4a². Next, let's look for all the ab terms: -10ab and 6ab. Combining these, we get -10ab + 6ab = -4ab. Finally, let's find the b² terms: 8b² and -9b². Combining these, we have 8b² - 9b² = -b². Now, we put all the simplified terms together to get our simplified expression: -4a² - 4ab - b². Congratulations! You've simplified the expression. The simplified form is much easier to work with than the original one, right? Understanding how to combine like terms is fundamental to simplifying almost any algebraic expression. The trick is to identify those terms with the same variables and then add or subtract their coefficients. If a term has no visible coefficient, like the b², remember it has an implied coefficient of 1. Always pay close attention to the signs. They make a huge difference, so make sure to carry them along with each term.

Now, let's move on to the next step, which is evaluating the expression by substituting the given values for a and b. This part is quite simple, because we know the value for a and b. It's all about plugging those values into the expression. Let's get to it!

Substituting Values and Finding the Answer

Now comes the fun part: evaluating the expression. We've simplified it, and now we're going to find its numerical value. We know that a = 1.1 and b = 0.8. We substitute these values into our simplified expression: -4a² - 4ab - b². First, let's substitute a and b: -4(1.1)² - 4(1.1)(0.8) - (0.8)². Now, we just need to do some simple arithmetic. Let's start by calculating the squares: (1.1)² = 1.21 and (0.8)² = 0.64. Our expression now becomes: -4(1.21) - 4(1.1)(0.8) - 0.64. Next, let's calculate the multiplication: -4 * 1.21 = -4.84 and 4 * 1.1 * 0.8 = 3.52. Our expression is now: -4.84 - 3.52 - 0.64. Finally, add all the numbers: -4.84 - 3.52 - 0.64 = -9.0. So, the value of the expression when a = 1.1 and b = 0.8 is -9.0. Fantastic! We've successfully simplified the expression, substituted the values, and arrived at the final answer. This kind of problem often appears in exams and tests. And now you have the tools to do well on them.

This entire process, from distributing to simplifying and evaluating, demonstrates the power of algebra. It's about using symbols and rules to solve problems and understand relationships. By practicing these steps, you'll become more confident in your ability to solve algebraic expressions. Just remember to take it step by step, be careful with the signs, and double-check your work. You've got this! And remember, practice makes perfect. The more you work through these types of problems, the easier and more natural they will become. Keep up the great work, and you will do great in your algebraic journey!