Factoring 2x² - 6x - 36 A Step-by-Step Guide

Factoring quadratic expressions is a fundamental skill in algebra. It allows us to simplify expressions, solve equations, and gain a deeper understanding of mathematical relationships. In this comprehensive guide, we will walk you through the process of completely factoring the quadratic expression 2x² - 6x - 36. We'll break down each step, providing explanations and insights to help you master this essential technique. By the end of this article, you'll be able to confidently tackle similar factoring problems and apply your knowledge to more advanced algebraic concepts. So, let's dive in and unlock the secrets of factoring!

Understanding Factoring

Before we begin, let's establish a clear understanding of what factoring means. In essence, factoring is the reverse process of expanding or multiplying expressions. When we factor an expression, we are breaking it down into its constituent parts, typically smaller expressions that, when multiplied together, yield the original expression. For instance, consider the number 12. We can factor it into 3 and 4 because 3 * 4 = 12. Similarly, in algebra, we can factor expressions involving variables and coefficients.

The goal of factoring completely is to break down an expression into its simplest possible factors. This means that each factor cannot be factored further. Factoring completely is crucial for various mathematical operations, such as simplifying fractions, solving equations, and analyzing graphs. It's a skill that builds the foundation for more advanced algebraic concepts.

When dealing with quadratic expressions, which are expressions of the form ax² + bx + c, where a, b, and c are constants, factoring often involves finding two binomials (expressions with two terms) that, when multiplied together, give us the original quadratic expression. The process may seem daunting at first, but with a systematic approach and practice, it becomes a manageable and rewarding skill.

Now that we have a solid grasp of what factoring entails, let's move on to the specific steps involved in factoring the expression 2x² - 6x - 36. We'll start by identifying the greatest common factor (GCF), a crucial first step in simplifying the expression and making it easier to factor.

Step 1: Identify the Greatest Common Factor (GCF)

The first step in factoring completely is to identify the greatest common factor (GCF) of all the terms in the expression. The GCF is the largest factor that divides evenly into each term. Finding the GCF simplifies the expression and makes the subsequent factoring steps easier. In our case, the expression is 2x² - 6x - 36. Let's examine the coefficients and the variable terms:

• The coefficients are 2, -6, and -36.
• The variable terms are x² and x.

To find the GCF of the coefficients, we look for the largest number that divides evenly into 2, -6, and -36. That number is 2. Next, we consider the variable terms. Both terms contain x, but the lowest power of x is x¹ (or simply x). Therefore, the GCF of the variable terms is x.

Combining the GCF of the coefficients and the variable terms, we find that the GCF of the entire expression 2x² - 6x - 36 is 2. Now, we can factor out the GCF from the expression.

Factoring out the GCF involves dividing each term in the expression by the GCF and writing the GCF outside a set of parentheses. In our case, we divide each term in 2x² - 6x - 36 by 2:

• 2x² / 2 = x²
• -6x / 2 = -3x
• -36 / 2 = -18

Now, we rewrite the expression with the GCF factored out:

2(x² - 3x - 18)

Factoring out the GCF has simplified our expression. We now have a simpler quadratic expression inside the parentheses, which is easier to factor than the original expression. In the next step, we will focus on factoring the quadratic expression x² - 3x - 18.

Step 2: Factor the Quadratic Expression

Now that we've factored out the GCF, we're left with the quadratic expression x² - 3x - 18. Our next task is to factor this expression into two binomials. To do this, we need to find two numbers that satisfy two conditions:

1. They multiply to give the constant term (-18).
2. They add up to give the coefficient of the x term (-3).

Let's systematically list the pairs of factors of -18 and see which pair adds up to -3:

• 1 and -18 (sum: -17)
• -1 and 18 (sum: 17)
• 2 and -9 (sum: -7)
• -2 and 9 (sum: 7)
• 3 and -6 (sum: -3) This is the pair we're looking for!
• -3 and 6 (sum: 3)

We found that the numbers 3 and -6 satisfy both conditions. They multiply to -18 and add up to -3. Now, we can use these numbers to write the factored form of the quadratic expression:

x² - 3x - 18 = (x + 3)(x - 6)

This means that when we multiply (x + 3) and (x - 6), we get x² - 3x - 18. We have successfully factored the quadratic expression!

Now that we've factored the expression inside the parentheses, we need to remember the GCF we factored out in the first step. We'll incorporate the GCF into our final factored form in the next step.

Step 3: Write the Complete Factored Form

We're in the final stretch of factoring completely! We've successfully factored out the GCF and factored the quadratic expression. Now, we need to combine these results to write the complete factored form of the original expression, 2x² - 6x - 36.

Recall that we factored out a GCF of 2 in the first step, which gave us:

2(x² - 3x - 18)

Then, we factored the quadratic expression x² - 3x - 18 into (x + 3)(x - 6). Now, we simply substitute the factored form of the quadratic expression back into the expression with the GCF:

2(x² - 3x - 18) = 2(x + 3)(x - 6)

Therefore, the completely factored form of 2x² - 6x - 36 is 2(x + 3)(x - 6). This is our final answer!

To ensure our answer is correct, we can multiply the factors together and see if we get the original expression. Let's do that:

2(x + 3)(x - 6) = 2(x² - 6x + 3x - 18) = 2(x² - 3x - 18) = 2x² - 6x - 36

The result matches our original expression, so we can be confident that our factoring is correct.

Step 4: Verify the Solution

To be absolutely certain of our answer, let's expand the factored form we obtained, 2(x + 3)(x - 6), and see if it matches the original expression, 2x² - 6x - 36. This step is crucial for verifying the accuracy of our factoring and ensuring that we haven't made any mistakes along the way.

Expanding the factored form involves multiplying the factors together. We'll start by multiplying the two binomials (x + 3) and (x - 6):

(x + 3)(x - 6) = x(x - 6) + 3(x - 6) = x² - 6x + 3x - 18 = x² - 3x - 18

Now, we multiply the result by the GCF, which is 2:

2(x² - 3x - 18) = 2x² - 6x - 36

The expanded form, 2x² - 6x - 36, is exactly the same as our original expression. This confirms that our factored form, 2(x + 3)(x - 6), is indeed the correct factorization of the given quadratic expression.

Verifying the solution is an essential step in the factoring process. It provides us with the assurance that our answer is accurate and that we have successfully factored the expression completely. By expanding the factored form and comparing it to the original expression, we can catch any potential errors and make necessary corrections. This practice not only enhances our problem-solving skills but also deepens our understanding of the relationship between factored and expanded forms of algebraic expressions.

Conclusion

In this comprehensive guide, we've walked through the process of factoring completely the quadratic expression 2x² - 6x - 36. We broke down the process into manageable steps, starting with identifying the greatest common factor (GCF), then factoring the remaining quadratic expression, and finally, writing the complete factored form. We also emphasized the importance of verifying the solution to ensure accuracy.

Factoring is a fundamental skill in algebra, and mastering it opens the door to solving a wide range of mathematical problems. It's a technique that you'll encounter frequently in various areas of mathematics, from solving equations to simplifying expressions and analyzing graphs. By understanding the steps involved and practicing consistently, you can become proficient in factoring and confidently tackle more complex algebraic challenges.

Remember, the key to success in factoring is to approach each problem systematically, break it down into smaller steps, and pay close attention to the details. With practice and perseverance, you'll develop a strong intuition for factoring and be able to apply this skill effectively in your mathematical journey. So, keep practicing, keep exploring, and keep expanding your knowledge of algebra!

The completely factored form of 2x² - 6x - 36 is 2(x + 3)(x - 6). This corresponds to option B in the original question.