Factoring 2x^2 - 4x + 160 A Step-by-Step Guide

In mathematics, factoring is a fundamental skill, especially when dealing with polynomial expressions. Factoring involves breaking down a polynomial into a product of simpler polynomials or factors. This process is crucial for solving equations, simplifying expressions, and understanding the behavior of functions. In this article, we will delve into the process of factoring the quadratic expression $2x^2 - 4x + 160$. We will explore the steps involved, common techniques, and provide a detailed explanation to ensure clarity and understanding. The given quadratic expression is $2x^2 - 4x + 160$. Our goal is to express this quadratic as a product of its factors, if possible. Factoring is a technique used to simplify complex expressions and make them easier to work with. The process involves identifying common factors and rewriting the expression as a product of these factors. In the context of quadratic expressions, factoring can help in solving equations, finding roots, and simplifying algebraic manipulations. Factoring is not just a mathematical exercise; it's a powerful tool with practical applications in various fields such as engineering, physics, computer science, and economics. The ability to factor expressions accurately and efficiently can significantly enhance problem-solving skills and is essential for advanced mathematical studies. By mastering factoring techniques, one can tackle more complex algebraic problems with confidence and precision. Understanding the underlying principles and methods of factoring provides a solid foundation for further mathematical explorations and real-world applications.

Step-by-Step Factoring Process

1. Identify Common Factors

The first step in factoring any polynomial expression is to identify common factors among all the terms. In our expression, $2x^2 - 4x + 160$, we can see that each term is divisible by 2. Factoring out the common factor simplifies the expression and makes it easier to work with. By recognizing and extracting the greatest common divisor (GCD), we reduce the complexity of the expression and set the stage for further factoring techniques. This initial step is crucial because it often reveals the underlying structure of the polynomial and facilitates subsequent steps in the factoring process. Neglecting to factor out common factors at the beginning can lead to more complicated calculations and potentially incorrect results. Therefore, always begin by checking for common factors to streamline the factoring process and enhance accuracy. In this case, the common factor of 2 simplifies the expression significantly, paving the way for more manageable factoring techniques.

So, we factor out 2:

$2(x^2 - 2x + 80)$

2. Analyze the Quadratic Expression

After factoring out the common factor, we are left with the quadratic expression $x^2 - 2x + 80$. Now, we need to analyze this expression to determine if it can be factored further. A quadratic expression in the form $ax^2 + bx + c$ can be factored into two binomials $(x + p)(x + q)$ if we can find two numbers, $p$ and $q$, such that their product equals $c$ and their sum equals $b$. In our case, $a = 1$, $b = -2$, and $c = 80$. The discriminant, calculated as $b^2 - 4ac$, provides valuable information about the nature of the roots and the factorability of the quadratic. If the discriminant is positive, the quadratic has two distinct real roots and can be factored into real binomials. If the discriminant is zero, the quadratic has one real root (a repeated root) and can be expressed as a perfect square. If the discriminant is negative, the quadratic has complex roots and cannot be factored using real numbers. Analyzing the quadratic expression involves a systematic approach to understanding its components and characteristics. This step is crucial for deciding the appropriate factoring method and ensuring an accurate result. Understanding the nature of the quadratic allows us to choose the most efficient and effective factoring technique.

3. Check for Factor Pairs

We need to find two numbers that multiply to 80 and add up to -2. Let's list the factor pairs of 80:

• 1 and 80
• 2 and 40
• 4 and 20
• 5 and 16
• 8 and 10

None of these pairs (or their negative counterparts) add up to -2. This indicates that the quadratic expression $x^2 - 2x + 80$ cannot be factored into simple binomials with integer coefficients. The absence of suitable factor pairs suggests that the roots of the quadratic equation are either irrational or complex. In such cases, attempting to factor the expression using traditional methods will not yield integer solutions. This step highlights the importance of systematically checking all possible factor pairs to ensure that no combination is overlooked. If no integer factor pairs meet the required conditions, it is necessary to consider alternative methods or conclude that the quadratic expression is not factorable over integers. This process of elimination is crucial in determining the most appropriate course of action for solving or simplifying the quadratic expression. Therefore, a thorough search for factor pairs is a key step in the factoring process.

4. Verify with the Discriminant

To confirm whether the quadratic expression can be factored using real numbers, we can use the discriminant, which is given by the formula:

$D = b^2 - 4ac$

In our case, $a = 1$, $b = -2$, and $c = 80$. Plugging these values into the formula, we get:

$D = (-2)^2 - 4(1)(80) = 4 - 320 = -316$

Since the discriminant is negative (-316), the quadratic equation $x^2 - 2x + 80 = 0$ has complex roots, meaning it cannot be factored using real numbers. The discriminant serves as a critical indicator of the nature of the roots of a quadratic equation. A negative discriminant confirms that the roots are complex conjugates, and the quadratic expression cannot be factored into real binomials. This information is crucial for determining the appropriate approach to solving quadratic equations or simplifying expressions. When the discriminant is negative, alternative methods such as the quadratic formula are required to find the complex roots. Understanding the implications of the discriminant allows mathematicians and students to efficiently solve quadratic equations and analyze their solutions. Therefore, the discriminant is a valuable tool in the analysis and manipulation of quadratic expressions.

Final Answer

Given that the quadratic expression $x^2 - 2x + 80$ cannot be factored further using real numbers, the factored form of the original expression $2x^2 - 4x + 160$ is:

$2(x^2 - 2x + 80)$

Discussion of the Options

Now, let's review the given options and see which one matches our result:

• $2(x+10)(x-8)$: Expanding this gives $2(x^2 + 2x - 80)$, which is not equal to $2x^2 - 4x + 160$.
• $2(x+10)(x+8)$: Expanding this gives $2(x^2 + 18x + 80)$, which is also not equal to $2x^2 - 4x + 160$.
• $2(x-10)(x+8)$: Expanding this gives $2(x^2 - 2x - 80)$, which is not equal to $2x^2 - 4x + 160$.
• $-2(x^2 - 2x + 80)$: This option incorrectly includes a negative sign outside the parentheses. While the expression inside the parentheses is correct, the overall expression does not match the original.

None of the provided options correctly factor the given quadratic expression. The correct representation remains $2(x^2 - 2x + 80)$, indicating that the quadratic part cannot be factored further using real numbers. This analysis underscores the importance of accurately factoring expressions and verifying the results through expansion. Each step in the factoring process must be carefully executed to avoid errors and ensure that the final expression is equivalent to the original. By systematically evaluating the given options, we confirm that the correct answer is the unfactorable form $2(x^2 - 2x + 80)$, which highlights the critical nature of precise mathematical manipulation.

Conclusion

In summary, when asked to factor the expression $2x^2 - 4x + 160$, we first identified and factored out the common factor of 2, resulting in $2(x^2 - 2x + 80)$. We then analyzed the quadratic expression $x^2 - 2x + 80$ to determine if it could be factored further. By checking for factor pairs and using the discriminant, we found that it could not be factored using real numbers due to the negative discriminant. Therefore, the final factored form of the expression is $2(x^2 - 2x + 80)$. Factoring quadratic expressions is a fundamental skill in algebra, with applications in various mathematical contexts and real-world problems. The ability to systematically factor expressions enables efficient simplification, solution-finding, and analysis of mathematical relationships. Understanding the underlying principles and techniques of factoring enhances problem-solving skills and provides a solid foundation for advanced mathematical studies. Mastery of factoring techniques, including identifying common factors, analyzing quadratic expressions, and using the discriminant, empowers individuals to tackle complex algebraic problems with confidence and precision. Therefore, the importance of thorough understanding and practice in factoring cannot be overstated for both academic and practical applications.