Solving Algebraic Equations A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little algebraic problem. We've got this expression: $3x(x-2) - 6x(x-3) - 5x(x+2) + 8x^2$ and we need to simplify it. Don't worry, it looks more intimidating than it actually is. We'll break it down step-by-step so everyone can follow along.

Understanding the Problem

Before we jump into the solution, let's make sure we understand what we're dealing with. This expression involves polynomials, specifically terms with the variable 'x' raised to different powers. Our goal is to simplify this expression by combining like terms. That means we'll be using the distributive property and then adding or subtracting terms with the same power of 'x'.

The expression can be a bit daunting at first glance, but fear not! The key to solving these types of problems is to take it one step at a time. We'll use the distributive property to get rid of the parentheses and then carefully combine the like terms. This methodical approach will make the whole process much more manageable and less prone to errors. Remember, practice makes perfect! So, the more you work through these types of problems, the more comfortable and confident you'll become.

Step-by-Step Solution

Let's start by applying the distributive property to each term:

1. Distribute the terms:

• $3x(x-2) = 3x^2 - 6x$
• $-6x(x-3) = -6x^2 + 18x$ (Remember to distribute the negative sign!)
• $-5x(x+2) = -5x^2 - 10x$

Now, let's rewrite the entire expression with these expanded terms:

$3x^2 - 6x - 6x^2 + 18x - 5x^2 - 10x + 8x^2$

2. Combine like terms:

Now, we need to group the terms with the same power of 'x'. We have terms with $x^2$ and terms with $x$. Let's rearrange the expression to make it clearer:

$(3x^2 - 6x^2 - 5x^2 + 8x^2) + (-6x + 18x - 10x)$

Now, let's add the coefficients of the $x^2$ terms:

$3 - 6 - 5 + 8 = 0$

So, the $x^2$ terms sum up to $0x^2$, which is just 0. That's pretty cool, right?

Next, let's add the coefficients of the $x$ terms:

$-6 + 18 - 10 = 2$

So, the $x$ terms sum up to $2x$.

3. The simplified expression:

Putting it all together, our simplified expression is:

$0 + 2x = 2x$

Identifying the Correct Answer

Looking at the options provided, we can see that the correct answer is:

d) $2x$

Common Mistakes and How to Avoid Them

Algebraic simplification can be tricky, and it's easy to make mistakes if you're not careful. Let's look at some common pitfalls and how to dodge them.

1. Forgetting the Distributive Property

The distributive property is your best friend in these situations. It's the key to unlocking the expression trapped inside those parentheses. Make sure you multiply each term inside the parentheses by the term outside. A common mistake is to only multiply the first term, which leads to incorrect simplification. Remember, it's like sharing – everyone inside the parentheses gets a piece!

2. Sign Errors

Sign errors are super sneaky and can trip you up if you're not vigilant. Pay close attention to the signs (positive and negative) when you're distributing, especially when there's a negative sign outside the parentheses. A negative multiplied by a negative becomes a positive, and a negative multiplied by a positive becomes a negative. Double-check your signs at every step to make sure everything's in order.

3. Combining Unlike Terms

You can only combine like terms, meaning terms with the same variable raised to the same power. For example, you can combine $3x^2$ and $-5x^2$ because they both have $x^2$, but you can't combine $3x^2$ and $2x$ because one has $x^2$ and the other has $x$. It's like trying to add apples and oranges – they're both fruit, but you can't say you have 5 apple-oranges!

4. Arithmetic Errors

Sometimes the simplest errors are the ones that get us. Arithmetic errors, like adding or subtracting numbers incorrectly, can throw off your entire solution. Take your time, double-check your calculations, and maybe even use a calculator for the trickier parts. It's better to be safe than sorry!

5. Not Simplifying Completely

Make sure you simplify the expression completely. This means combining all like terms and making sure there are no further simplifications possible. Leaving the expression partially simplified means you haven't finished the job. Think of it like cleaning your room – you can't just make the bed and call it done, you have to put everything away!

How to Avoid These Mistakes

• Write Clearly: Neatness counts! Write each step clearly and legibly so you can easily follow your work and spot any mistakes. It's like having a roadmap for your solution.
• Show Your Work: Don't try to do everything in your head. Show each step of your calculation, so you can easily go back and check your work if you need to. It's like leaving a trail of breadcrumbs so you can find your way back.
• Double-Check: Always double-check your work, especially your signs and arithmetic. It's like proofreading a paper before you submit it – you want to catch any errors before someone else does.
• Practice, Practice, Practice: The more you practice, the more comfortable you'll become with algebraic simplification, and the fewer mistakes you'll make. It's like learning to ride a bike – the more you practice, the better you get.

Real-World Applications of Algebraic Simplification

You might be thinking,