Factoring $5x^2 - 24x + 27$: A Step-by-Step Guide

Are you struggling with factoring quadratic expressions? Don't worry, you're not alone! Factoring can seem tricky at first, but with a little practice and the right approach, you'll be a pro in no time. In this guide, we'll break down the process of factoring the quadratic expression $5x^2 - 24x + 27$ step by step. We'll cover the key concepts, walk through the calculations, and provide helpful tips to make factoring easier. So, let's dive in and unlock the secrets of factoring!

Understanding Quadratic Expressions

Before we jump into the factoring process, let's make sure we're all on the same page about quadratic expressions. A quadratic expression is a polynomial expression of the form $ax^2 + bx + c$, where a, b, and c are constants (numbers) and x is a variable. The coefficient of the $x^2$ term is a, the coefficient of the x term is b, and the constant term is c. In our example, $5x^2 - 24x + 27$, we have a = 5, b = -24, and c = 27.

Factoring a quadratic expression means rewriting it as a product of two binomials (expressions with two terms). For example, factoring $x^2 + 5x + 6$ gives us $(x + 2)(x + 3)$. When we multiply these binomials together, we get back the original quadratic expression. Factoring is a crucial skill in algebra, as it helps us solve equations, simplify expressions, and understand the behavior of functions. In this article, we will learn how to factor a quadratic expression completely. This means we will break down the expression into its simplest factors, ensuring that no further factoring is possible. Factoring completely is essential for solving equations and simplifying expressions effectively, giving you a solid foundation in algebra.

The AC Method: A Powerful Factoring Technique

There are several methods for factoring quadratic expressions, but one of the most reliable and widely used is the AC method. This method is particularly helpful when the coefficient of the $x^2$ term (a) is not equal to 1, as in our example. The AC method involves the following steps:

1. Multiply a and c: Calculate the product of the coefficient of the $x^2$ term (a) and the constant term (c). This product is often referred to as the AC value.
2. Find two factors of AC that add up to b: Identify two numbers that multiply to give you the AC value and add up to the coefficient of the x term (b). This is the most crucial step, and it may require some trial and error.
3. Rewrite the middle term: Replace the middle term (bx) with the sum of two terms, using the factors you found in the previous step as coefficients of x.
4. Factor by grouping: Group the first two terms and the last two terms of the expression. Factor out the greatest common factor (GCF) from each group.
5. Factor out the common binomial: If you've done everything correctly, you'll have a common binomial factor in both groups. Factor out this common binomial to obtain the factored form of the quadratic expression.

Let's apply the AC method to our example, $5x^2 - 24x + 27$, to see how it works in practice.

Applying the AC Method to $5x^2 - 24x + 27$

Now, let's walk through the steps of the AC method to factor our quadratic expression, $5x^2 - 24x + 27$. This detailed walkthrough will help you understand each step and build confidence in applying the AC method to other quadratic expressions.

Step 1: Multiply a and c

First, we need to find the product of a and c. In our expression, a = 5 and c = 27. So,

• AC = 5 * 27 = 135

Step 2: Find Two Factors of AC that Add Up to b

This is the most critical step. We need to find two numbers that multiply to 135 and add up to b, which is -24. Since the product is positive and the sum is negative, both factors must be negative. Let's list the pairs of factors of 135 and see which pair adds up to -24:

• -1 and -135 (sum = -136)
• -3 and -45 (sum = -48)
• -5 and -27 (sum = -32)
• -9 and -15 (sum = -24)

Bingo! We found our factors: -9 and -15. These two numbers multiply to 135 and add up to -24.

Step 3: Rewrite the Middle Term

Now, we'll rewrite the middle term, -24x, using the factors we just found. We'll replace -24x with -9x - 15x. Our expression now looks like this:

• $5x^2 - 9x - 15x + 27$

Step 4: Factor by Grouping

Next, we'll group the first two terms and the last two terms and factor out the greatest common factor (GCF) from each group:

• $(5x^2 - 9x) + (-15x + 27)$

From the first group, the GCF is x, so we factor it out:

• $x(5x - 9)$

From the second group, the GCF is -3, so we factor it out (note the negative sign, which is crucial for the next step):

• $-3(5x - 9)$

Now our expression looks like this:

• $x(5x - 9) - 3(5x - 9)$

Step 5: Factor Out the Common Binomial

Notice that we now have a common binomial factor, $(5x - 9)$, in both terms. We can factor this out:

• $(5x - 9)(x - 3)$

And that's it! We've factored the quadratic expression $5x^2 - 24x + 27$ completely. The factored form is $(5x - 9)(x - 3)$.

Checking Your Work

It's always a good idea to check your factoring to make sure you've done it correctly. To check, simply multiply the binomials you obtained in the factored form. If you get back the original quadratic expression, you've factored correctly.

Let's check our work:

• $(5x - 9)(x - 3) = 5x(x - 3) - 9(x - 3)$
• $= 5x^2 - 15x - 9x + 27$
• $= 5x^2 - 24x + 27$

Great! We got back the original expression, so our factoring is correct.

Tips and Tricks for Factoring

Factoring can be challenging, but here are some tips and tricks to make the process smoother:

• Always look for a GCF first: Before applying any other factoring method, check if there's a greatest common factor that can be factored out of all the terms. This simplifies the expression and makes it easier to factor further.
• Practice makes perfect: The more you practice factoring, the better you'll become at recognizing patterns and applying the appropriate techniques. Work through a variety of examples to build your skills.
• Don't be afraid to guess and check: If you're not sure which factors to use, try different combinations until you find the right one. With practice, you'll develop a better intuition for which factors are likely to work.
• Use online tools and calculators: There are many online resources available to help you check your factoring or get hints if you're stuck. These tools can be valuable learning aids, but remember to focus on understanding the process rather than just getting the answer.
• Understand the signs: Pay close attention to the signs of the coefficients and the constant term. This will help you determine the signs of the factors you need to find.
• Stay organized: Keep your work neat and organized. Write down each step clearly, so you can easily follow your reasoning and spot any mistakes.

By following these tips and tricks, you'll be well on your way to mastering factoring quadratic expressions.

Conclusion

Factoring quadratic expressions is a fundamental skill in algebra, and the AC method is a powerful tool for tackling these problems, especially when the leading coefficient isn't 1. By understanding the steps involved and practicing consistently, you can confidently factor a wide range of quadratic expressions. Remember to always check your work and utilize the tips and tricks discussed to improve your efficiency and accuracy. With dedication and the right approach, you'll master factoring and unlock new possibilities in your mathematical journey. In conclusion, mastering the AC method and consistently practicing will make factoring quadratic expressions like $5x^2 - 24x + 27$ much easier. Keep practicing, and you'll soon find yourself solving these problems with confidence. For more resources on factoring and other algebraic topics, consider visiting websites like Khan Academy which offers comprehensive lessons and practice exercises.