Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of simplifying algebraic expressions. This is a fundamental skill in algebra, and it's super important for everything from solving equations to understanding more complex mathematical concepts. We're going to break down the expression $\frac{30 x^2-90 x}{80 x}$ step by step, so you can see exactly how it works. Don't worry, it's not as scary as it looks! We'll go through everything, making sure you grasp the key ideas and techniques involved. By the end, you'll be able to tackle similar problems with confidence. So, grab your pencils and let's get started. Simplifying expressions is all about making them as easy to understand and work with as possible. This usually means reducing the expression to its simplest form by canceling out common factors, combining like terms, and performing the indicated operations. It's like cleaning up a messy room – you're just organizing and streamlining things! Remember, the goal is always to get the expression into a form where it's easier to see what's going on and easier to use in further calculations. Let's look at the given options to see what we're working with. A) $x+2$, B) $\frac{1}{x+6}$, C) $\frac{8}{3(x-3)}$, and D) $\frac{3(x-3)}{8}$. These represent possible simplified forms of the original expression. Our job is to find which one is correct. We'll start with the original expression and begin the simplification process.

Breaking Down the Expression: The First Steps

Okay, so the expression we need to simplify is $\frac{30 x^2-90 x}{80 x}$. The first thing we want to do is look for common factors in the numerator (the top part of the fraction). Notice that both terms in the numerator, $30x^2$ and $-90x$, have a common factor of $30x$. We can factor out $30x$ from the numerator. This is like finding the biggest number or expression that divides evenly into all the terms. In this case, both $30x^2$ and $90x$ are divisible by $30x$. When we factor out $30x$, we are essentially dividing each term by $30x$. Let's do it:

• $30x^2$ divided by $30x$ gives us $x$
• $-90x$ divided by $30x$ gives us $-3$

So, after factoring out $30x$ from the numerator, our expression becomes $30x(x - 3)$. Now, the expression looks like this: $\frac{30x(x - 3)}{80x}$. See, we've already made it a bit simpler! Factoring is a crucial skill because it helps us uncover hidden relationships and simplifies complex expressions. Always start by looking for those common factors. Once you get the hang of it, you'll be surprised at how much easier it makes the problem. Remember, we are trying to rewrite the expression in a simpler form. Now, the expression $\frac{30x(x - 3)}{80x}$ will be easier to simplify further. The next step is to look for common factors between the numerator and the denominator. The numerator is now in factored form, which is great because it makes it easier to see common factors. The denominator is $80x$. Now we have $30x$ in the numerator and $80x$ in the denominator. Both have an $x$ term, but we can also simplify the numbers 30 and 80. Let's see how.

Simplifying the Expression Further

Now, we have $\frac{30x(x - 3)}{80x}$. Our next move is to cancel out common factors between the numerator and the denominator. Notice that both the numerator and the denominator have $x$. We can cancel these out, provided that $x$ is not equal to zero (because division by zero is undefined). Also, we can simplify the fraction $\frac{30}{80}$ by dividing both the numerator and denominator by their greatest common divisor, which is 10. Dividing 30 by 10 gives us 3, and dividing 80 by 10 gives us 8. So, $\frac{30}{80}$ simplifies to $\frac{3}{8}$. Now that we've taken care of the $x$ terms and simplified the fraction, our expression transforms into $\frac{3(x - 3)}{8}$. We've simplified the original expression $\frac{30 x^2-90 x}{80 x}$ step by step, first by factoring out the common factor of $30x$ from the numerator, resulting in $\frac{30x(x - 3)}{80x}$. We then canceled out the common factor of $x$ from the numerator and denominator, and simplified the fraction $\frac{30}{80}$ to $\frac{3}{8}$. This left us with our final simplified expression, $\frac{3(x - 3)}{8}$.

Matching with the Options and the Final Answer

Alright, we have simplified the original expression to $\frac{3(x - 3)}{8}$. Now, let's look back at the answer choices provided:

A) $x + 2$ B) $\frac{1}{x + 6}$ C) $\frac{8}{3(x - 3)}$ D) $\frac{3(x - 3)}{8}$

By comparing our simplified expression with the options, we can see that our answer, $\frac{3(x - 3)}{8}$, perfectly matches option D. So, D is the correct answer! Nice work! Recognizing and applying simplification techniques is super important when solving mathematical problems, and it will save you a lot of time and effort in the long run. Congratulations on successfully simplifying the expression! Keep practicing, and you'll become a pro at these problems in no time. Always remember the fundamental steps: factor where possible, cancel out common terms, and simplify numerical fractions. Practicing these skills regularly will significantly improve your ability to solve a wide variety of algebraic problems.

Recap and Key Takeaways

Let's quickly recap what we did:

1. Factored the numerator: We identified and factored out the common factor $30x$ from the numerator, transforming the expression to $\frac{30x(x - 3)}{80x}$.
2. Canceled common factors: We canceled out the common factor $x$ from the numerator and the denominator.
3. Simplified the fraction: We simplified the fraction $\frac{30}{80}$ to $\frac{3}{8}$.
4. Final result: We arrived at the simplified expression $\frac{3(x - 3)}{8}$.

These are the core steps to simplifying algebraic expressions like the one we've just worked through. Remember that simplification is all about making an expression easier to understand and use. And there you have it, the simplified form of our algebraic expression! I hope you guys found this breakdown helpful and that you feel more confident in tackling similar problems. Simplifying expressions might seem tricky at first, but with practice, you'll get the hang of it. Remember to always look for common factors, and don't forget to simplify any fractions. Good luck with your future math endeavors, and keep practicing!