Algebraic Expression Simplification

Hey everyone! Today, we're diving deep into the awesome world of algebraic expressions. You know, those cool combinations of numbers, variables, and operations that pop up everywhere in math and science. We're going to tackle a specific problem: simplifying the expression (a-2)(a-2)-a(a-3). Now, this might look a little intimidating at first glance, but trust me, guys, once you break it down, it's totally manageable and even kinda fun. This process is all about making complex expressions simpler and easier to understand, which is a super important skill in algebra. We'll walk through each step, explaining why we do what we do, so you can get a solid grasp on how to handle similar problems in the future. Think of this as your ultimate cheat sheet for mastering algebraic simplification. We'll be using fundamental algebraic principles like the distributive property and combining like terms, so pay close attention! By the end of this, you'll be able to confidently simplify this expression and others like it. Let's get this party started!

Understanding the Expression: (a-2)(a-2)-a(a-3)

Alright guys, let's first get a good look at the expression we're working with: (a-2)(a-2)-a(a-3). The goal here is to rewrite this entire thing as a single polynomial, which basically means an expression with one or more terms, where each term is a constant multiplied by one or more variables raised to non-negative integer powers. See that (a-2)(a-2) part? That's a perfect square. Remember squaring something means multiplying it by itself. So, (a-2)(a-2) is the same as (a-2) squared. We can expand this using the FOIL method (First, Outer, Inner, Last) or by remembering the formula for a binomial squared: (x-y)² = x² - 2xy + y². In our case, 'x' is 'a' and 'y' is '2'. So, (a-2)² becomes a² - 2(a)(2) + 2², which simplifies to a² - 4a + 4. Pretty neat, right? Now, let's look at the second part of the expression: -a(a-3). This is where the distributive property comes into play. We need to multiply '-a' by each term inside the parentheses. So, -a * a gives us -a², and -a * -3 gives us +3a. So, -a(a-3) expands to -a² + 3a. See how we handle each part separately? This makes the whole process much less chaotic. The key takeaway here is to recognize the different algebraic structures within the expression and know which rules to apply to each part. It's like having different tools for different jobs in your algebraic toolbox.

Step 1: Expand the First Part - (a-2)(a-2)

Let's really focus on the first chunk of our expression: (a-2)(a-2). As we mentioned, this is (a-2) squared. We can tackle this in a couple of ways, and both lead to the same awesome result. The FOIL method is a classic for multiplying two binomials. It stands for First, Outer, Inner, Last.

• First: Multiply the first terms in each binomial: a * a = a².
• Outer: Multiply the outer terms: a * (-2) = -2a.
• Inner: Multiply the inner terms: (-2) * a = -2a.
• Last: Multiply the last terms: (-2) * (-2) = +4.

Now, we add all these results together: a² + (-2a) + (-2a) + 4. Combining the like terms (-2a and -2a), we get a² - 4a + 4.

Alternatively, we can use the binomial square formula: (x - y)² = x² - 2xy + y². Here, 'x' is 'a' and 'y' is '2'. Plugging these values in, we get:

• x²: a²
• -2xy: -2 * a * 2 = -4a
• +y²: +2² = +4

So, again, we arrive at a² - 4a + 4. Both methods confirm that (a-2)(a-2) beautifully expands into a² - 4a + 4. It's great to have multiple ways to solve a problem; it really solidifies your understanding, right guys? This expansion is the first major step in transforming our original expression into a polynomial.

Step 2: Expand the Second Part - a(a-3)

Now, let's shift our focus to the second part of the original expression: -a(a-3). This part involves the distributive property, which is a fundamental concept in algebra. The distributive property states that for any numbers a, b, and c, a(b + c) = ab + ac. In our case, we have -a multiplying the binomial (a - 3). So, we need to distribute the -a to both terms inside the parentheses.

• Multiply -a by the first term, a: (-a) * a = -a².
• Multiply -a by the second term, -3: (-a) * (-3) = +3a.

Putting it all together, the expansion of -a(a-3) is -a² + 3a.

It's super important to pay attention to the signs here, guys. Remember that multiplying two negative numbers results in a positive number. That's why -a * -3 gives us a positive 3a. This careful handling of signs is crucial for accuracy in algebraic manipulations. This step simplifies the second component of our expression, preparing us for the final combination.

Step 3: Combine the Expanded Parts

Okay, guys, we've successfully expanded both parts of our original expression. We found that (a-2)(a-2) expands to a² - 4a + 4, and -a(a-3) expands to -a² + 3a. Now, we need to put these pieces back together according to the original expression, which was (a-2)(a-2) - a(a-3). So, we substitute our expanded forms:

(a² - 4a + 4) + (-a² + 3a)

Notice that we've kept the expanded forms in parentheses for clarity, especially the second one which had a negative sign in front of it in the original problem. The next crucial step is to remove the parentheses and then combine like terms. Since we are adding the two expanded expressions, the signs inside the parentheses don't change when we remove them:

a² - 4a + 4 - a² + 3a

Now for the fun part: combining like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have:

• Terms with a²: a² and -a²
• Terms with a: -4a and +3a
• Constant terms: +4

Let's combine them:

• a² - a² = 0 (These cancel each other out!)
• -4a + 3a = -a
• The constant term +4 remains as it is.

So, when we combine all these, we get 0 - a + 4, which simplifies to -a + 4.

This is our final polynomial form! It's much simpler than the original expression, and that's the beauty of algebraic simplification. We've successfully navigated the expansions and cancellations to arrive at a concise answer. High five!

Step 4: Final Polynomial Form

And there you have it, folks! After all the expanding and combining, the expression (a-2)(a-2)-a(a-3) has been successfully transformed into its simplest polynomial form. We meticulously expanded the first part, (a-2)(a-2), using either the FOIL method or the binomial square formula, resulting in a² - 4a + 4. Then, we applied the distributive property to the second part, -a(a-3), which yielded -a² + 3a. Finally, we brought these expanded pieces together and combined the like terms. The a² terms cancelled each other out (a² - a² = 0), the a terms combined to -a (-4a + 3a = -a), and the constant term +4 remained. Therefore, the simplified polynomial expression is -a + 4.

This is the ultimate goal of this kind of problem: to take a complex-looking expression and reduce it to its most basic and understandable polynomial form. It's a testament to the power of algebraic rules and properties like the distributive property and the concept of combining like terms. Remember, guys, mastering these steps isn't just about solving this one problem; it's about building a strong foundation in algebra that will help you tackle even more challenging mathematical concepts down the line. Keep practicing, and you'll be an algebra whiz in no time! You've got this!