Simplify (6x^2 + 20) / 4: A Step-by-Step Guide

Hey guys! Today, we're diving into a super common algebra problem: simplifying the expression $\frac{6x^2 + 20}{4}$. Don't worry; it's much easier than it looks. We'll break it down step-by-step, so you'll be simplifying like a pro in no time. Let's get started!

Understanding the Basics

Before we jump into the problem, let's make sure we're all on the same page with some basic math concepts. Simplifying an expression means rewriting it in a simpler, more manageable form. In this case, we want to make the fraction $\frac{6x^2 + 20}{4}$ as clean and straightforward as possible. This involves looking for common factors and reducing the expression. Think of it like decluttering – we're getting rid of the unnecessary stuff to make the important parts shine.

When we talk about fractions, remember that the top part is the numerator (in our case, $6x^2 + 20$) and the bottom part is the denominator (which is 4). Simplifying fractions often involves finding the greatest common divisor (GCD) between the numerator and the denominator and then dividing both by that GCD. However, with expressions like this, we can also use the distributive property to simplify each term individually.

Why is simplifying important? Well, simplified expressions are easier to work with in further calculations, help in understanding the relationship between variables, and can make complex problems much more manageable. Plus, it's a fundamental skill in algebra, so mastering it will definitely help you out in the long run.

Step-by-Step Simplification

Okay, let's tackle the expression $\frac{6x^2 + 20}{4}$. Here’s how we can simplify it:

Step 1: Distribute the Denominator

The first thing we can do is to think of the fraction as a division applied to each term in the numerator. So, we can rewrite the expression as:

$\frac{6x^2}{4} + \frac{20}{4}$

This step uses the property that $\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}$. It's like splitting the fraction into two separate fractions, each with the same denominator.

Step 2: Simplify Each Fraction

Now, let's simplify each fraction individually. For the first fraction, $\frac{6x^2}{4}$, we can simplify the numerical part. Both 6 and 4 are divisible by 2. So, we divide both the numerator and the denominator by 2:

$\frac{6x^2}{4} = \frac{6 ÷ 2}{4 ÷ 2} x^2 = \frac{3}{2}x^2$

For the second fraction, $\frac{20}{4}$, this is a simple division:

$\frac{20}{4} = 5$

Step 3: Combine the Simplified Terms

Now that we've simplified both fractions, we combine them back together:

$\frac{3}{2}x^2 + 5$

So, the simplified form of $\frac{6x^2 + 20}{4}$ is $\frac{3}{2}x^2 + 5$.

Alternative Method: Factoring

Another way to approach this problem is by factoring out a common factor from the numerator before dividing. Let's try that:

Step 1: Factor out the Greatest Common Factor (GCF)

Look at the terms in the numerator, $6x^2$ and $20$. What's the largest number that divides both 6 and 20? That would be 2. So, we factor out 2 from the numerator:

$6x^2 + 20 = 2(3x^2 + 10)$

Step 2: Rewrite the Expression

Now, rewrite the original expression with the factored numerator:

$\frac{6x^2 + 20}{4} = \frac{2(3x^2 + 10)}{4}$

Step 3: Simplify the Fraction

Notice that we now have a factor of 2 in both the numerator and the denominator. We can simplify the fraction by dividing both by 2:

$\frac{2(3x^2 + 10)}{4} = \frac{2 ÷ 2}{4 ÷ 2} (3x^2 + 10) = \frac{1}{2} (3x^2 + 10)$

Step 4: Distribute (if needed)

We can leave the expression as $\frac{1}{2} (3x^2 + 10)$ or distribute the $\frac{1}{2}$ to get rid of the parentheses:

$\frac{1}{2} (3x^2 + 10) = \frac{3}{2}x^2 + \frac{10}{2} = \frac{3}{2}x^2 + 5$

As you can see, both methods lead us to the same simplified expression: $\frac{3}{2}x^2 + 5$.

Common Mistakes to Avoid

When simplifying expressions like this, there are a few common pitfalls you should watch out for:

• Forgetting to distribute: Make sure you apply the division to every term in the numerator. It's easy to simplify one term and forget about the others.
• Incorrectly simplifying fractions: Double-check your division. A small mistake in simplifying a fraction can throw off the entire problem.
• Not factoring completely: If you choose to factor, make sure you factor out the greatest common factor. Otherwise, you might have to simplify further down the line.
• Mixing up operations: Remember the order of operations (PEMDAS/BODMAS). Simplify within parentheses first, then exponents, then multiplication and division, and finally addition and subtraction.

Practice Problems

Want to test your skills? Try simplifying these expressions:

1. $\frac{8x^2 + 12}{4}$
2. $\frac{10x^3 + 15}{5}$
3. $\frac{9x^2 + 21}{3}$

Solutions:

1. $\frac{8x^2 + 12}{4} = 2x^2 + 3$
2. $\frac{10x^3 + 15}{5} = 2x^3 + 3$
3. $\frac{9x^2 + 21}{3} = 3x^2 + 7$

Conclusion

Simplifying expressions like $\frac{6x^2 + 20}{4}$ might seem daunting at first, but with a step-by-step approach and a bit of practice, you'll become a master at it. Remember to distribute the denominator, simplify each fraction, and watch out for common mistakes. Whether you prefer distributing or factoring, the key is to break the problem down into smaller, more manageable steps. Keep practicing, and you'll be simplifying algebraic expressions like a total boss! You got this! Keep up the amazing work, and you'll master it in no time! Don't give up, and you'll see great results.