Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of algebraic expressions. We'll be tackling a problem that involves simplifying a complex expression, a common task in algebra. Specifically, we're looking to simplify the expression: $\frac{x^2+10 x+25}{x+5}-\frac{x^2-6}{x-5}$. Don't worry, it might look a little intimidating at first, but we'll break it down step-by-step to make it super clear and easy to understand. Let's get started, shall we?

Understanding the Problem: The Core Concepts

Alright guys, before we jump into the nitty-gritty of the solution, let's make sure we're all on the same page. The heart of this problem lies in simplifying algebraic expressions involving fractions. Remember that when we're dealing with fractions, especially in algebra, we often need to find a common denominator to combine them. This is because we can't directly add or subtract fractions unless they share the same denominator. It's like comparing apples and oranges – you need a common unit (the denominator) to accurately compare them. Another crucial concept here is factoring. We'll be using factoring to simplify the expression, by breaking down polynomials into their simpler forms. Factoring helps us to identify common factors that can be cancelled out, further simplifying the expression. It is important to know the basics of how to handle polynomials. Polynomials are the foundation of this question, and the more you know about them the better you will understand the questions. So, keeping these basics in mind, let's explore the key steps to solve this problem.

Now, let's look at the given expression: $\frac{x^2+10 x+25}{x+5}-\frac{x^2-6}{x-5}$. Our goal is to find an equivalent expression from the given options: A. $\frac{2 x^2-19}{x-5}$, B. $\frac{2 x^2-19}{x^2-25}$, C. $\frac{19}{x-5}$, D. $\frac{-19}{x-5}$. To solve this, we'll need to combine the two fractions. Because the denominators are different, we can't directly subtract them. We will need to manipulate both the numerator and the denominator of each fraction so that they share a common denominator. The first step will be to find this common denominator. The common denominator will then be used to add the fractions, and the combined expression will be simplified to one of the given answers.

Step-by-Step Solution: Cracking the Code

Alright, let's get down to business! Here's how we're going to solve this algebraic expression step by step. First things first: Factoring the Numerators. Notice that the first numerator, $x^2 + 10x + 25$, looks familiar. It's a perfect square trinomial! It factors into $(x + 5)^2$ or $(x+5)(x+5)$. This is a super handy trick to remember, as it can often make the expression easier to work with. The second numerator, $x^2 - 6$, doesn't factor neatly, so we'll leave it as is for now. Now we need to Identify the Common Denominator. Looking at our two fractions, we have $(x+5)$ and $(x-5)$ as the denominators. Since they don't share any common factors, the common denominator will be their product, which is $(x+5)(x-5)$ or $x^2 - 25$. Got it? Now we'll Rewrite the Fractions with the Common Denominator. To do this, we'll multiply the numerator and denominator of each fraction by the factor it's missing to get the common denominator. So, the first fraction $\frac{x^2+10 x+25}{x+5}$ becomes $\frac{(x+5)(x+5)(x-5)}{(x+5)(x-5)}$. The second fraction, $\frac{x^2-6}{x-5}$ becomes $\frac{(x^2-6)(x+5)}{(x-5)(x+5)}$.

Next, Simplify the Numerators. After multiplying the numerators, the first fraction will be $(x+5)(x+5)(x-5)$, this simplifies to $(x^2+10x+25)(x-5)$, which equals to $x^3 + 5x^2 -25x -125$. The second fraction's numerator is $x^2 - 6$ and $x + 5$, therefore, $(x^2-6)(x+5)$ becomes $x^3 + 5x^2 - 6x - 30$. Now we can Subtract the Fractions. Subtracting the second fraction from the first, we get $\frac{x^3 + 5x^2 -25x -125 - (x^3 + 5x^2 - 6x - 30)}{x^2 - 25}$. This simplifies further to $\frac{x^3 + 5x^2 -25x -125 - x^3 - 5x^2 + 6x + 30}{x^2 - 25}$. Combining like terms, we arrive at $\frac{-19x - 95}{x^2 - 25}$. The last step is Simplifying the Expression. We can factor out a -19 from the numerator, so we have $\frac{-19(x + 5)}{(x+5)(x-5)}$. Now we can cancel out the $(x + 5)$ terms. This leaves us with $\frac{-19}{x-5}$. Boom! We've simplified the expression.

The Answer and Why It Matters

So, guys, after all that work, what's the answer? The expression equivalent to $\frac{x^2+10 x+25}{x+5}-\frac{x^2-6}{x-5}$ is D. $\frac{-19}{x-5}$. Congratulations if you got it right! If not, don't sweat it. The beauty of math is that with practice, it becomes easier. Remember, each step is critical, from finding the common denominator to simplifying the final result. Understanding how to simplify algebraic expressions is a fundamental skill in algebra. It helps us solve equations, analyze functions, and understand more complex mathematical concepts. Furthermore, these skills are not just confined to math class. They're incredibly useful in fields like computer science, engineering, and even economics, where manipulating and simplifying complex formulas is essential. Being able to break down a complex problem into smaller, manageable steps is a skill that will serve you well in many aspects of life, not just in mathematics. Now, let's explore the key takeaways from our problem.

Key Takeaways and Tips for Success

• Master the Basics: A strong grasp of factoring, fraction operations, and polynomial manipulation is key. Make sure you're comfortable with these concepts before tackling more complex problems. Regularly practice these concepts to build fluency.
• Take it Slow: Don't rush! Algebraic simplification requires precision. Go step-by-step, double-check your work, and don't be afraid to take your time. Rushing leads to careless errors, so accuracy is much more important than speed.
• Look for Patterns: Recognizing patterns, like perfect square trinomials or the difference of squares, can save you time and effort. Keep an eye out for these patterns as you solve problems. They can be huge time savers.
• Practice, Practice, Practice: The more you practice, the better you'll become. Work through different types of problems to solidify your understanding. Get in the habit of doing practice problems frequently to keep your skills sharp.
• Don't Be Afraid to Ask for Help: If you get stuck, don't hesitate to ask your teacher, classmates, or online resources for help. Learning from others is an important part of the process.

And there you have it! We've successfully simplified a complex algebraic expression. Keep practicing, stay curious, and you'll become a math whiz in no time. See ya next time!