Factoring Expressions A Step-by-Step Guide To Solving 4(x^2-2x)-2(x^2-3)

Hey guys! Today, we're diving deep into the fascinating world of factoring expressions. We're going to break down a common problem type you might encounter in algebra and show you exactly how to solve it step-by-step. Our focus is on a specific expression: $4(x^2-2x)-2(x^2-3)$. We'll explore how to transform this into its factored form and discuss the underlying principles that make it all click. Whether you're a student grappling with homework, a teacher looking for clear explanations, or just someone who loves math, this guide is for you!

Understanding the Expression

Before we jump into the solution, let's make sure we understand what we're dealing with. The expression $4(x^2-2x)-2(x^2-3)$ looks a bit intimidating at first glance, but it's really just a combination of terms involving the variable x. The key here is to recognize the structure of the expression. We have two main parts, each involving a quadratic term ($x^2$) and a linear term (x). These parts are multiplied by constants (4 and -2), and then subtracted from each other. To simplify and factor this, we'll need to use the distributive property and combine like terms. This initial assessment is crucial because it sets the stage for our strategy. We're not just blindly applying formulas; we're understanding the anatomy of the expression, which will help us choose the right tools and techniques for the job. Keep in mind that mastering this initial understanding is super helpful for more complex problems down the road.

Step 1 Distribute the Constants

Our first move is to apply the distributive property to get rid of those parentheses. This means we'll multiply the constants outside the parentheses by each term inside. For the first part, $4(x^2-2x)$, we multiply 4 by both $x^2$ and -2x. This gives us $4x^2 - 8x$. Similarly, for the second part, $-2(x^2-3)$, we multiply -2 by both $x^2$ and -3. Remember to pay close attention to the signs! -2 times $x^2$ is $-2x^2$, and -2 times -3 is +6. So, this part becomes $-2x^2 + 6$. Now, our expression looks like this: $4x^2 - 8x - 2x^2 + 6$. See? It's already starting to look a bit simpler! This step is all about expanding the expression and making it easier to see which terms we can combine. Distribution is a fundamental technique in algebra, so getting comfortable with it is a big win. By breaking down the problem into smaller, manageable steps, we're making the whole process less daunting and more approachable.

Step 2 Combine Like Terms

Now that we've distributed the constants, it's time to combine like terms. Like terms are those that have the same variable raised to the same power. In our expression, $4x^2 - 8x - 2x^2 + 6$, we have two terms with $x^2$ ($4x^2$ and $-2x^2$) and one term with x (-8x) and a constant term (6). Let's combine the $x^2$ terms first: $4x^2 - 2x^2$ equals $2x^2$. Now, our expression looks like $2x^2 - 8x + 6$. The -8x term stays as it is because there are no other x terms to combine with, and the constant +6 also remains unchanged. Combining like terms is a crucial simplification step. It reduces the number of terms in the expression, making it cleaner and easier to work with. Think of it like organizing your desk – putting similar items together makes everything more manageable. By this point, we've transformed the original expression into a much simpler quadratic expression, setting us up nicely for the next step: factoring.

Step 3 Factor out the Greatest Common Factor (GCF)

Before we jump into the more complex factoring techniques, let's look for the greatest common factor (GCF). The GCF is the largest number or expression that divides evenly into all terms. In our simplified expression, $2x^2 - 8x + 6$, we can see that each term is divisible by 2. So, 2 is our GCF. We factor out the 2 by dividing each term by 2 and placing it outside a set of parentheses. This gives us $2(x^2 - 4x + 3)$. Factoring out the GCF is a smart move because it simplifies the quadratic expression inside the parentheses, making it easier to factor further. It's like taking out the big pieces of the puzzle first, so the remaining pieces fit together more easily. This step also helps us avoid dealing with larger numbers in the subsequent factoring steps, which can reduce the chances of making mistakes. So, always be on the lookout for a GCF – it's a real time-saver!

Step 4 Factor the Quadratic Expression

Now comes the fun part: factoring the quadratic expression inside the parentheses, $x^2 - 4x + 3$. This is a classic quadratic expression in the form of $ax^2 + bx + c$, where a = 1, b = -4, and c = 3. To factor this, we're looking for two numbers that multiply to c (3) and add up to b (-4). Think of it like a little number puzzle! The numbers that fit the bill are -1 and -3 because (-1) * (-3) = 3 and (-1) + (-3) = -4. Once we find these numbers, we can rewrite the quadratic expression in its factored form as $(x - 1)(x - 3)$. So, our expression now looks like $2(x - 1)(x - 3)$. Factoring quadratics is a fundamental skill in algebra, and mastering it opens doors to solving a wide range of problems. There are different techniques for factoring, but the one we used here – finding two numbers that multiply to c and add to b – is particularly useful when a = 1. Practice makes perfect with this skill, so keep at it!

The Final Factored Form

We've reached the finish line! Our original expression, $4(x^2-2x)-2(x^2-3)$, has been transformed into its factored form: $2(x - 1)(x - 3)$. This is the answer we were looking for! Let's recap the steps we took to get here: 1. We distributed the constants. 2. We combined like terms. 3. We factored out the greatest common factor (GCF). 4. We factored the remaining quadratic expression. Each step played a crucial role in simplifying the expression and revealing its factored form. Factoring is a powerful tool in algebra. It allows us to rewrite expressions in a more compact and useful way. Factored forms are particularly handy when solving equations, simplifying fractions, and analyzing functions. So, mastering factoring is an investment that pays off big time in your mathematical journey. Good job, guys!

Identifying the Correct Option

Now that we've successfully factored the expression, let's circle back to the original question and identify the correct answer choice. We were given four options: A. $2(x-1)(x-3)$ B. $(2x-3)(x+1)$ C. $2(x+1)(x+3)$ D. $(2x+3)(x+1)$ Our factored form, $2(x - 1)(x - 3)$, matches option A perfectly! This step is a crucial reminder to always double-check your work and make sure you're selecting the answer that truly matches your solution. It's easy to make a small mistake along the way, so taking a moment to compare your answer with the options provided can save you from losing points on a test or assignment. Plus, it reinforces the sense of accomplishment when you see that your hard work has paid off. So, always take that extra step to confirm your answer – it's the cherry on top of a successful problem-solving effort!

Key Takeaways and Factoring Tips

Alright, let's solidify our understanding with some key takeaways and factoring tips! First off, remember the importance of the distributive property. It's your go-to tool for expanding expressions and getting rid of parentheses. Next, always combine like terms to simplify the expression before attempting to factor. This makes the factoring process much smoother. Don't forget to look for the greatest common factor (GCF) – factoring it out early can save you headaches later. When factoring quadratic expressions, remember the