Factor $2x^2 + 3x - 54$: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of factoring quadratic expressions. Specifically, we're going to tackle the expression $2x^2 + 3x - 54$. Factoring quadratics might seem daunting at first, but with a systematic approach, it becomes much easier. This article will not only help you understand how to factor this particular expression but also equip you with the skills to handle similar problems. So, let's break it down step by step and find those factors!

Understanding Quadratic Expressions

Before we jump into the factoring process, let's quickly recap what a quadratic expression is. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (in this case, x) is 2. The general form of a quadratic expression is $ax^2 + bx + c$, where a, b, and c are constants. In our case, for the expression $2x^2 + 3x - 54$, a = 2, b = 3, and c = -54. Understanding these coefficients is crucial for the factoring process.

Why is factoring important? Factoring is a fundamental skill in algebra and has various applications. It helps in solving quadratic equations, simplifying expressions, and understanding the behavior of quadratic functions. When you factor a quadratic expression, you're essentially rewriting it as a product of two binomials. These binomials are the factors of the quadratic expression. For example, if we can factor $2x^2 + 3x - 54$ into (something) * (something else), those “somethings” are the factors we're looking for. Factoring is also a reverse process of expanding, where expanding a product of binomials gives us the quadratic expression. The connection between factoring and expanding also provides a way to check our work. After factoring, we can always expand the factors to verify that they multiply back to the original quadratic expression. This check ensures accuracy and gives you confidence in your solution.

Now, let's get into the factoring process. We'll use a method that works well for quadratics where the leading coefficient (the a value) is not 1. It might seem a bit more involved than simpler factoring problems, but it's a reliable method that you can use consistently. Understanding the different techniques for factoring can significantly enhance your problem-solving skills. For instance, recognizing patterns such as the difference of squares or perfect square trinomials can lead to faster solutions. However, for general quadratic expressions, a systematic approach like the one we're about to use is often the most effective.

The Factoring Process: A Step-by-Step Guide

So, how do we factor $2x^2 + 3x - 54$? We'll use a method that involves finding two numbers that satisfy specific conditions related to the coefficients of the quadratic expression. Here’s the breakdown:

1. Multiply a and c: First, we multiply the coefficient of the $x^2$ term (a) by the constant term (c). In our case, this is 2 * (-54) = -108. This step is crucial because it sets the stage for finding the right combination of factors. The product ac gives us the target number we need to work with in the next step. The sign of this product is also important, as it tells us whether we're looking for two numbers with the same sign (both positive or both negative) or two numbers with different signs. In this case, the negative product indicates that we need two numbers with different signs.

2. Find Two Numbers: Next, we need to find two numbers that multiply to -108 (the result from step 1) and add up to 3 (the coefficient of the x term, b). This is the heart of the factoring process. It might involve some trial and error, but a systematic approach can make it easier. We start by listing pairs of factors of -108 and checking their sums. Since the product is negative, we know one number must be positive and the other negative. We need to find a pair where the difference between the numbers is 3 (since the b value is 3). Let's consider some pairs:

   • 1 and -108 (sum = -107)
   • -1 and 108 (sum = 107)
   • 2 and -54 (sum = -52)
   • -2 and 54 (sum = 52)
   • 3 and -36 (sum = -33)
   • -3 and 36 (sum = 33)
   • 4 and -27 (sum = -23)
   • -4 and 27 (sum = 23)
   • 6 and -18 (sum = -12)
   • -6 and 18 (sum = 12)
   • 9 and -12 (sum = -3)
   • -9 and 12 (sum = 3)

   Aha! We found our pair: -9 and 12. These numbers multiply to -108 and add up to 3. Finding this pair is a critical step. Without it, we cannot proceed with factoring the quadratic expression. In more complex cases, you might need to systematically test different factor pairs or use prime factorization to help identify the right numbers. The more you practice, the quicker you'll become at this step.

3. Rewrite the Middle Term: Now, we rewrite the middle term (3x) using the two numbers we found (-9 and 12). So, we replace 3x with -9x + 12x. The expression becomes: $2x^2 - 9x + 12x - 54$. This step might seem a bit strange, but it's the key to factoring by grouping, which is the next step. By splitting the middle term, we've created a four-term expression that we can factor more easily.

   It's important to note that the order in which you write the terms (-9x and 12x) doesn't matter. You'll get the same result either way. However, sometimes one order might make the factoring by grouping step slightly easier. If you find yourself struggling with the next step, you can always try switching the order of these terms.

4. Factor by Grouping: We now have four terms, and we'll factor by grouping. We group the first two terms and the last two terms: $(2x^2 - 9x) + (12x - 54)$. Next, we factor out the greatest common factor (GCF) from each group. From the first group, the GCF is x, and from the second group, the GCF is 6. Factoring these out, we get: x(2x - 9) + 6(2x - 9). Notice that we now have a common binomial factor: (2x - 9). This is a good sign! It means we're on the right track. Factoring by grouping is a powerful technique that allows us to handle more complex quadratic expressions by breaking them down into smaller, more manageable parts.

5. Factor out the Common Binomial: Finally, we factor out the common binomial factor (2x - 9) from the entire expression. This gives us: (2x - 9)(x + 6). And there you have it! We've factored the quadratic expression $2x^2 + 3x - 54$ into (2x - 9)(x + 6). This is the final step in the factoring process, where we combine the results of the previous steps to express the original quadratic as a product of two binomials. Each binomial represents a factor of the quadratic expression.

Identifying the Correct Factors

Now that we've factored the expression, let's look back at the original question. We needed to select two options that are factors of $2x^2 + 3x - 54$. Based on our factoring, the factors are (2x - 9) and (x + 6). So, the correct options are:

• $2x - 9$
• $x + 6$

It’s always a good idea to double-check your work, especially in math. We can expand our factored form (2x - 9)(x + 6) to see if we get back the original expression:

(2x - 9)(x + 6) = 2x(x) + 2x(6) - 9(x) - 9(6) = $2x^2 + 12x - 9x - 54 = 2x^2 + 3x - 54$

Yep, it checks out! We've successfully factored the quadratic expression and identified the correct factors. Checking your answer by expanding the factors is a crucial step in ensuring the accuracy of your solution. It's a simple process that can save you from making mistakes, especially in exams or when solving more complex problems that rely on correct factoring.

Tips and Tricks for Factoring Quadratics

Factoring quadratics can become second nature with practice. Here are a few extra tips and tricks to help you along the way:

• Always look for a GCF first: Before you start the factoring process, check if there's a greatest common factor that can be factored out of all the terms. This simplifies the expression and makes the factoring process easier. For example, if you had the expression $4x^2 + 6x - 108$, you could factor out a 2 first, resulting in $2(2x^2 + 3x - 54)$.
• Practice makes perfect: The more you practice factoring, the quicker and more confident you'll become. Try working through a variety of examples, including those with different coefficients and signs. There are many online resources and textbooks that offer practice problems for factoring quadratics. Working through these problems will help you develop a better understanding of the process and improve your speed and accuracy.
• Use the quadratic formula: If you're having trouble factoring a quadratic expression, you can always use the quadratic formula to find the roots. The roots can then be used to write the factors. The quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of whether it can be factored easily or not. It provides a reliable method for finding the roots, which can then be used to determine the factors.
• Check your work: As we showed earlier, always check your factored form by expanding it to make sure it matches the original expression. This helps you catch any errors and ensures that your answer is correct. Checking your work is a good habit to develop in mathematics, as it can prevent mistakes and improve your overall accuracy.

Conclusion

So, there you have it! We've walked through the process of factoring the quadratic expression $2x^2 + 3x - 54$ step by step. We found the factors to be (2x - 9) and (x + 6). Remember, factoring quadratics is a skill that improves with practice. By understanding the process and using the tips we've discussed, you'll be able to tackle even more challenging problems. Keep practicing, and you'll become a factoring pro in no time! Factoring quadratics is not just a mathematical exercise; it's a fundamental skill that has applications in various fields, including engineering, physics, and computer science. Mastering this skill will open doors to more advanced mathematical concepts and real-world problem-solving.