Factor Completely 2x^2 - 16x + 30 Step-by-Step Solution

In this article, we will delve into the process of completely factoring the quadratic expression 2x^2 - 16x + 30. Factoring quadratic expressions is a fundamental skill in algebra, crucial for solving equations, simplifying expressions, and understanding the behavior of polynomial functions. We will break down each step, providing a clear and detailed explanation to ensure you grasp the underlying concepts. By the end of this guide, you will be able to confidently factor similar expressions and apply these techniques to more complex problems. Let's embark on this algebraic journey and unlock the secrets of factoring!

Understanding the Basics of Factoring

Before we dive into the specific example of factoring 2x^2 - 16x + 30, it's important to establish a solid foundation in the basics of factoring. Factoring, in essence, is the reverse process of expansion or distribution. When we expand an expression, we multiply terms together; when we factor, we decompose an expression into its constituent factors, which, when multiplied, yield the original expression. Think of it as breaking down a number into its prime factors, but now we're dealing with algebraic expressions. This involves identifying common factors, recognizing patterns, and applying various techniques to rewrite the expression in a more simplified, factored form. A strong understanding of these principles is the bedrock for mastering more advanced factoring techniques.

What is Factoring?

Factoring is the process of breaking down an algebraic expression into a product of simpler expressions. These simpler expressions are called factors. For example, the number 12 can be factored into 2 × 2 × 3, where 2 and 3 are prime factors. Similarly, in algebra, we can factor expressions like polynomials into simpler polynomials or monomials. The goal is to rewrite the expression as a product of its factors, which can then be used to solve equations, simplify expressions, or analyze functions. This decomposition into factors offers a different perspective on the original expression, often revealing hidden properties and relationships. Understanding the factors of an expression is akin to understanding the building blocks of a structure – it gives you a deeper insight into its nature and behavior.

Why is Factoring Important?

Factoring is a cornerstone of algebra and has far-reaching applications in mathematics and related fields. One of the primary reasons factoring is so important is its role in solving equations. Many equations, particularly polynomial equations, can be solved by setting the expression equal to zero and then factoring. Once factored, the solutions (or roots) of the equation can be easily found by setting each factor equal to zero and solving for the variable. This technique is especially crucial for solving quadratic equations, which appear in various contexts, from physics to engineering to economics. Beyond solving equations, factoring is also essential for simplifying algebraic expressions. By factoring out common factors or recognizing special patterns, we can reduce complex expressions to their simplest forms, making them easier to work with and understand. This simplification is critical in many areas of mathematics, including calculus and linear algebra. Furthermore, factoring plays a vital role in understanding the behavior of functions. The factored form of a polynomial, for instance, reveals its roots (x-intercepts) and provides valuable information about its graph. In essence, factoring is a powerful tool that unlocks the underlying structure of algebraic expressions and equations, enabling us to solve problems, simplify complexities, and gain deeper insights.

Step-by-Step Factoring of 2x^2 - 16x + 30

Now, let's dive into the step-by-step process of factoring the given quadratic expression, 2x^2 - 16x + 30. We'll meticulously walk through each stage, ensuring clarity and understanding. Our goal is not just to arrive at the final factored form but to also comprehend the logic and techniques involved. This methodical approach will equip you with the ability to tackle similar factoring problems with confidence. Remember, factoring is a skill that improves with practice, so actively following along and applying these steps will be invaluable.

Step 1: Identifying the Greatest Common Factor (GCF)

The first step in factoring any polynomial is to identify and factor out the greatest common factor (GCF). The GCF is the largest factor that divides evenly into all the terms of the polynomial. In our expression, 2x^2 - 16x + 30, we observe the coefficients 2, -16, and 30. The GCF of these numbers is 2. Notice that the variable 'x' is not present in the constant term (30), so 'x' cannot be part of the GCF. Factoring out the GCF of 2, we rewrite the expression as follows:

2(x^2 - 8x + 15)

This step simplifies the expression inside the parentheses, making it easier to factor further. Factoring out the GCF is a crucial first step because it reduces the complexity of the expression and often reveals the underlying structure more clearly. It's like taking out the common building block to see what's left to work with. Neglecting to factor out the GCF can lead to more complicated factoring in later steps, so it's always a good practice to start here.

Step 2: Factoring the Quadratic Expression Inside the Parentheses

After factoring out the GCF, we are left with the quadratic expression inside the parentheses: x^2 - 8x + 15. This is a trinomial of the form ax^2 + bx + c, where a = 1, b = -8, and c = 15. To factor this quadratic expression, we need to find two numbers that multiply to 'c' (15) and add up to 'b' (-8). This is a common technique for factoring quadratics, and it relies on the relationship between the coefficients and the factors of the expression. Let's systematically find these two numbers.

We need two numbers that multiply to 15 and add to -8. Consider the factors of 15: 1 and 15, 3 and 5. Since the product is positive and the sum is negative, both numbers must be negative. Testing the pairs, we find that -3 and -5 satisfy both conditions:

(-3) * (-5) = 15

(-3) + (-5) = -8

Thus, we can rewrite the quadratic expression as a product of two binomials:

(x - 3)(x - 5)

This step is the heart of the factoring process, where we decompose the quadratic into its linear factors. The ability to identify the correct pair of numbers is crucial, and it often requires some trial and error. However, with practice, you'll develop an intuition for this process. Remember, the goal is to rewrite the quadratic as a product of two simpler expressions, which in this case are the binomials (x - 3) and (x - 5).

Step 3: Writing the Completely Factored Form

Now that we've factored the quadratic expression inside the parentheses, we need to combine this result with the GCF we factored out in the first step. This will give us the completely factored form of the original expression, 2x^2 - 16x + 30. We found that the GCF was 2, and the factored form of the quadratic expression x^2 - 8x + 15 is (x - 3)(x - 5). Combining these, we get the completely factored form:

2(x - 3)(x - 5)

This is the final answer, and it represents the original expression as a product of its irreducible factors. The completely factored form is not just a simplified representation; it also reveals important information about the expression, such as its roots (the values of x that make the expression equal to zero). In this case, the roots are x = 3 and x = 5. Writing the completely factored form is the culmination of the factoring process, and it's essential to present the answer in this form to ensure that the expression is factored as much as possible.

Verifying the Solution

Before finalizing our answer, it's crucial to verify that our factored form is indeed equivalent to the original expression. This step acts as a safety check, ensuring that we haven't made any errors during the factoring process. The verification process involves expanding the factored form and comparing it to the original expression. If the two expressions are identical, we can be confident in our solution. This verification step is a fundamental part of mathematical problem-solving, and it's particularly important in factoring, where errors can easily occur. Let's go through the expansion process step by step.

Expanding the Factored Form

To verify our solution, we need to expand the factored form, which is 2(x - 3)(x - 5). This involves multiplying the factors together and then distributing any remaining terms. We'll start by multiplying the two binomials (x - 3) and (x - 5) using the distributive property (often referred to as the FOIL method – First, Outer, Inner, Last):

(x - 3)(x - 5) = x(x - 5) - 3(x - 5)

= x^2 - 5x - 3x + 15

= x^2 - 8x + 15

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

2(x^2 - 8x + 15) = 2x^2 - 16x + 30

This expansion process systematically reverses the factoring steps, allowing us to see if we arrive back at the original expression. Each step in the expansion must be carefully executed to avoid errors. The distributive property is the key tool here, ensuring that each term in one factor is multiplied by each term in the other factor. Once the expansion is complete, we can compare the result to the original expression.

Comparing with the Original Expression

After expanding the factored form, we obtained the expression 2x^2 - 16x + 30. Now, we need to compare this with the original expression, which was also 2x^2 - 16x + 30. As we can see, the expanded form is exactly the same as the original expression. This confirms that our factoring is correct, and we can be confident in our solution.

The act of comparing the expanded form with the original expression is a critical step in the verification process. It's not enough to just perform the expansion; we must also explicitly check that the two expressions match. This comparison helps us catch any potential errors that might have occurred during the factoring or expansion steps. If the expressions do not match, it indicates that there was a mistake, and we need to revisit the factoring process to identify and correct the error. In this case, since the expressions match, we have successfully verified our solution.

Conclusion

In conclusion, we have successfully factored the quadratic expression 2x^2 - 16x + 30 completely. We followed a systematic approach, which included identifying and factoring out the greatest common factor (GCF), factoring the resulting quadratic expression, and finally, verifying our solution by expanding the factored form. The completely factored form of the expression is:

2(x - 3)(x - 5)

This process demonstrates the importance of understanding the fundamentals of factoring and applying them methodically. Factoring is a crucial skill in algebra, with applications in various areas of mathematics and beyond. By mastering these techniques, you can simplify expressions, solve equations, and gain a deeper understanding of algebraic relationships. Remember, practice is key to developing proficiency in factoring, so continue to apply these methods to different problems to hone your skills.

This comprehensive guide has provided a step-by-step explanation of how to factor the given quadratic expression. By following these steps and understanding the underlying principles, you can confidently tackle similar factoring problems in the future. Factoring is not just a mechanical process; it's a way of understanding the structure of algebraic expressions and their relationships. So, embrace the challenge, practice consistently, and unlock the power of factoring in your mathematical journey.

The correct answer is D. 2(x-5)(x-3).