Factoring Quadratics: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into the world of factoring quadratic expressions. Today, we're going to tackle the problem of factoring the expression $3x^2 - 36x + 60$. Factoring might seem a little tricky at first, but trust me, with a few simple steps, you'll be breaking down these expressions like a pro. This guide will walk you through the process, ensuring you understand each step. We'll examine the given options and choose the correct one. So, grab your pencils and let's get started! In this guide, we'll use bold and italics to emphasize important points. Make sure to pay attention because this is one of the most fundamental concepts to get a good score in your high school and college algebra classes.

Understanding the Basics of Factoring Quadratics

Before we jump into the problem, let's quickly recap what factoring is all about. Factoring a quadratic expression means rewriting it as a product of two binomials. A binomial is simply an expression with two terms, like $(x - 2)$ or $(x + 5)$. When we factor, we're essentially "undoing" the multiplication that would have been done to get the original quadratic expression. The general form of a quadratic expression is $ax^2 + bx + c$, where 'a', 'b', and 'c' are constants. Our goal is to find two binomials that, when multiplied together, give us the original expression. One of the very first things that you have to do is check for a Greatest Common Factor (GCF). If there is a common factor, factor it out before anything else. This makes the remaining expression easier to handle. Keep in mind that factoring is the reverse of expanding (or multiplying) out expressions. Understanding the relationship between these two processes can really help you get a better grasp of the subject. It is important to remember that there are several techniques for factoring. Different expressions may require different approaches. Practice is key! The more you factor, the more familiar you will become with the patterns and tricks involved. This is true for every topic in math. It is always a good idea to check your work by multiplying the factors back out to make sure you arrive at the original expression. This simple step can save you a lot of time and frustration in the long run. Also, note that some quadratic expressions are not factorable using real numbers. These are known as "prime" quadratics. However, in this case, we have a factorable expression.

Step-by-Step Factorization of $3x^2 - 36x + 60$

Now, let's get down to business and factor the expression $3x^2 - 36x + 60$. I'll show you step-by-step how to do it. The best way to learn math is by doing the problems! Here's how we'll break it down:

1. Find the Greatest Common Factor (GCF): Look at the coefficients of each term (3, -36, and 60). Is there a number that divides evenly into all of them? Yes! The GCF is 3. Let's factor that out:

$3x^2 - 36x + 60 = 3(x^2 - 12x + 20)$

We've simplified the expression, which makes it easier to work with. Always remember to look for a GCF first. It simplifies your job.

2. Factor the Remaining Quadratic: Now, we focus on factoring the quadratic expression inside the parentheses: $x^2 - 12x + 20$. We're looking for two numbers that multiply to give us 20 (the constant term) and add up to -12 (the coefficient of the x term). Think about the factors of 20: 1 and 20, 2 and 10, 4 and 5. Notice that -2 and -10 satisfy these conditions because -2 * -10 = 20 and -2 + -10 = -12. So, we can rewrite the quadratic as:

$x^2 - 12x + 20 = (x - 2)(x - 10)$

3. Combine the Factors: Remember the GCF we factored out in the first step? We need to include it in our final answer. So, the completely factored form of the original expression is:

$3x^2 - 36x + 60 = 3(x - 2)(x - 10)$

That's it! We've successfully factored the quadratic expression. It's often helpful to keep in mind the different types of factoring problems you might encounter, such as those where 'a' is equal to 1, or those where 'a' is not equal to 1. This helps you approach the problem using the correct method. This will help you get faster at solving this type of problem.

Examining the Answer Choices

Now that we have factored the expression, let's look at the given answer choices and see which one matches our solution:

a. $(x - 9)(x - 8)$ b. $3(x - 10)(x - 2)$ c. $(x - 6)(x + 10)$ d. $3(x - 10)(x + 2)$

As we derived, the correct factored form is $3(x - 10)(x - 2)$. This matches answer choice b. Let's quickly eliminate the other options to be absolutely sure:

• Option a: $(x - 9)(x - 8)$ does not have the GCF of 3 and it is not equivalent to the original equation.
• Option c: $(x - 6)(x + 10)$ does not have the GCF of 3 and the sign is wrong.
• Option d: $3(x - 10)(x + 2)$ has the correct GCF, but the signs are wrong.

So, the correct answer is indeed option b. Knowing how to effectively evaluate multiple-choice questions can be beneficial, especially when you are running short on time. Understanding the different question types and how to approach them can save you a lot of time. Also, you have to stay calm, don't get nervous, and always go back and check your work!

Tips for Factoring Success

Factoring can get easier with practice. Here are some tips to help you succeed. Try these tips the next time you factor and see how they can improve your results.

• Always Look for the GCF First: This is the golden rule. Factoring out the GCF simplifies the remaining quadratic and makes it easier to factor. This single step can make the difference between an easy problem and a difficult one.
• Practice Makes Perfect: The more you factor, the better you'll become at recognizing patterns and finding the correct factors. Do as many practice problems as you can. It helps to review different types of problems.
• Check Your Work: After factoring, always multiply the factors back together to ensure you get the original expression. This is a great way to catch any errors you might have made. This step is super important.
• Understand the Signs: Pay close attention to the signs (+ or -) in the original expression. They play a crucial role in determining the signs of the factors. This is where most students mess up.
• Use Different Techniques: Be familiar with various factoring techniques, such as factoring by grouping, using the quadratic formula, and the ac method. This will give you more tools to solve different types of problems.
• Stay Organized: Write down each step clearly and neatly. This will help you avoid making careless mistakes. This is a general skill that can apply to any topic.

Conclusion

So there you have it, guys! We've successfully factored the quadratic expression $3x^2 - 36x + 60$. Remember to take it step by step, look for the GCF, and practice, practice, practice! With these strategies, you'll be well on your way to mastering quadratic factorization. Factoring is a fundamental skill in algebra, and it opens the door to understanding more complex concepts. Remember, if you get stuck, don't be afraid to go back to the basics and review the concepts. With patience and persistence, you'll conquer any factoring problem that comes your way. Keep up the great work, and happy factoring!