Factoring Quadratic Expressions A Step-by-Step Solution For 2x² - 11x - 6
In the realm of algebra, quadratic expressions hold a significant place, appearing in various mathematical contexts and real-world applications. A quadratic expression is a polynomial of degree two, generally represented in the form ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. Mastering the art of manipulating and simplifying these expressions is crucial for success in algebra and beyond. One fundamental technique in this domain is factoring, which involves breaking down a quadratic expression into a product of two linear expressions. This skill is indispensable for solving quadratic equations, simplifying rational expressions, and tackling a wide range of algebraic problems.
In our exploration today, we'll delve into the intricacies of factoring the specific quadratic expression 2x² - 11x - 6. This expression serves as an excellent example to illustrate the standard methods employed in factoring quadratics, particularly those where the leading coefficient (the coefficient of the x² term) is not equal to 1. We'll walk through the process step-by-step, shedding light on the underlying logic and techniques involved. By the end of this discussion, you'll not only understand how to factor this particular expression but also gain a solid foundation for tackling other quadratic factoring challenges.
Understanding the nuances of factoring quadratic expressions goes beyond mere algebraic manipulation. It fosters critical thinking, problem-solving skills, and a deeper appreciation for the structure of mathematical expressions. As we embark on this journey, remember that practice is key. The more you engage with these types of problems, the more intuitive the factoring process will become. So, let's roll up our sleeves and unravel the mysteries of 2x² - 11x - 6!
The Factoring Challenge 2x² - 11x - 6
Our primary objective is to identify the equivalent expression for the given quadratic expression: 2x² - 11x - 6. This task involves factoring the quadratic, which means we aim to rewrite it as a product of two binomials. Factoring is the reverse process of expanding, where we multiply binomials to obtain a quadratic expression. In this case, we are looking for two binomials that, when multiplied together, yield 2x² - 11x - 6. The challenge lies in determining the correct combination of constants and variables within the binomials.
Several methods can be employed to factor quadratic expressions. One common technique is the trial-and-error method, where we systematically try different combinations of binomials until we find the ones that match the original expression. Another approach is the AC method, which involves finding two numbers that multiply to the product of the leading coefficient (a) and the constant term (c) and add up to the middle coefficient (b). These numbers are then used to rewrite the middle term, allowing us to factor by grouping. We'll be using a strategic combination of these methods to efficiently factor the given expression.
Before we dive into the factoring process, it's essential to grasp the significance of the coefficients in the quadratic expression. The leading coefficient (2 in this case) affects the possible combinations of terms in the binomials. The constant term (-6) provides clues about the signs and magnitudes of the constants within the binomials. And the middle coefficient (-11) dictates the sum of the products of the inner and outer terms of the binomials. By carefully analyzing these coefficients, we can narrow down the possibilities and streamline the factoring process.
Factoring quadratic expressions is a fundamental skill in algebra, with applications spanning various mathematical and scientific domains. Mastering this technique empowers us to solve equations, simplify expressions, and gain a deeper understanding of mathematical relationships. As we proceed, we'll not only focus on finding the correct factored form of 2x² - 11x - 6 but also emphasize the underlying principles and strategies that can be applied to a wide range of factoring problems.
Step-by-Step Factoring Process
To accurately factor the quadratic expression 2x² - 11x - 6, let's embark on a structured, step-by-step journey. This approach will not only lead us to the correct factored form but also illuminate the underlying principles of factoring quadratic expressions with a leading coefficient other than 1.
Step 1: Identify the Coefficients
The first step in factoring any quadratic expression is to identify the coefficients a, b, and c in the standard form ax² + bx + c. In our expression, 2x² - 11x - 6, we have:
- a = 2
- b = -11
- c = -6
These coefficients hold the key to unlocking the factored form of the expression. The value of a indicates that the leading terms in the binomial factors will likely involve 2x and x. The value of c suggests possible constant terms within the binomials, and the value of b gives us a target for the sum of the inner and outer products of the binomial terms.
Step 2: Apply the AC Method
The AC method is a powerful technique for factoring quadratic expressions, especially when the leading coefficient (a) is not equal to 1. This method involves the following steps:
- Calculate AC: Multiply the coefficients a and c. In our case, AC = 2 * (-6) = -12.
- Find Two Numbers: Identify two numbers that multiply to AC (-12) and add up to b (-11). These numbers are -12 and 1, because (-12) * 1 = -12 and (-12) + 1 = -11.
These two numbers, -12 and 1, are crucial because they will allow us to rewrite the middle term of the quadratic expression and factor by grouping.
Step 3: Rewrite the Middle Term
Using the two numbers we found in Step 2 (-12 and 1), we rewrite the middle term (-11x) of the quadratic expression as the sum of two terms:
- 2x² - 11x - 6 = 2x² - 12x + 1x - 6
We have essentially split the -11x term into -12x and +1x. This manipulation doesn't change the value of the expression but sets the stage for factoring by grouping.
Step 4: Factor by Grouping
Now, we group the first two terms and the last two terms of the rewritten expression:
- (2x² - 12x) + (1x - 6)
Next, we factor out the greatest common factor (GCF) from each group:
- 2x(x - 6) + 1(x - 6)
Notice that both groups now share a common binomial factor: (x - 6). We factor out this common binomial:
- (2x + 1)(x - 6)
And there you have it! We have successfully factored the quadratic expression 2x² - 11x - 6 into the product of two binomials: (2x + 1)(x - 6).
This step-by-step process demonstrates the power of the AC method in factoring quadratic expressions. By systematically breaking down the problem into smaller, manageable steps, we can confidently tackle even the most challenging factoring tasks.
Evaluating the Answer Choices
Now that we have factored the quadratic expression 2x² - 11x - 6 into (2x + 1)(x - 6), let's meticulously evaluate the given answer choices to pinpoint the correct equivalent expression. This process involves comparing our factored form with each of the provided options, ensuring that we select the one that perfectly matches our result. Understanding how to verify your answer is just as crucial as the factoring process itself.
A. (2x + 1)(x - 6)
This answer choice directly matches the factored form we obtained through the step-by-step process. To confirm its correctness, we can expand this expression using the FOIL method (First, Outer, Inner, Last) and see if it simplifies back to the original quadratic expression.
Expanding (2x + 1)(x - 6):
- First: (2x)(x) = 2x²
- Outer: (2x)(-6) = -12x
- Inner: (1)(x) = x
- Last: (1)(-6) = -6
Combining these terms, we get 2x² - 12x + x - 6, which simplifies to 2x² - 11x - 6. This perfectly matches our original expression, making option A the correct answer.
B. 2(x - 3)(x + 1)
To evaluate this option, we again expand the expression:
- First, expand (x - 3)(x + 1): x² + x - 3x - 3 = x² - 2x - 3
- Then, multiply by 2: 2(x² - 2x - 3) = 2x² - 4x - 6
This result, 2x² - 4x - 6, does not match our original expression 2x² - 11x - 6, so option B is incorrect.
C. (2x + 3)(x - 2)
Expanding this expression using the FOIL method:
- First: (2x)(x) = 2x²
- Outer: (2x)(-2) = -4x
- Inner: (3)(x) = 3x
- Last: (3)(-2) = -6
Combining these terms, we get 2x² - 4x + 3x - 6, which simplifies to 2x² - x - 6. This does not match the original expression, so option C is incorrect.
D. 2(x + 3)(x - 2)
Expanding this expression:
- First, expand (x + 3)(x - 2): x² - 2x + 3x - 6 = x² + x - 6
- Then, multiply by 2: 2(x² + x - 6) = 2x² + 2x - 12
This result, 2x² + 2x - 12, does not match our original expression 2x² - 11x - 6, making option D incorrect.
By systematically evaluating each answer choice, we have definitively confirmed that option A, (2x + 1)(x - 6), is the only expression equivalent to the given quadratic expression 2x² - 11x - 6.
Conclusion Correct Answer A (2x + 1)(x - 6)
In this comprehensive exploration, we successfully factored the quadratic expression 2x² - 11x - 6 and meticulously evaluated the given answer choices to arrive at the correct equivalent expression. Our journey began with a clear understanding of quadratic expressions and the fundamental concept of factoring. We then delved into a step-by-step factoring process, employing the AC method to rewrite the middle term and factor by grouping. This systematic approach not only led us to the factored form (2x + 1)(x - 6) but also provided valuable insights into the techniques applicable to a wider range of factoring problems.
Our evaluation of the answer choices further solidified our understanding of the factoring process. By expanding each option and comparing the result with the original expression, we were able to confidently confirm that option A, (2x + 1)(x - 6), is the only expression equivalent to 2x² - 11x - 6. This exercise highlights the importance of verifying your answer in mathematical problem-solving, ensuring accuracy and reinforcing your grasp of the underlying concepts.
Factoring quadratic expressions is a cornerstone of algebra, and mastering this skill opens doors to more advanced mathematical concepts. The techniques we've explored here, such as the AC method and factoring by grouping, are not only applicable to this specific problem but also serve as valuable tools in your mathematical arsenal. As you continue your journey in mathematics, remember that practice and persistence are key. The more you engage with factoring problems, the more intuitive the process will become.
Furthermore, the ability to factor quadratic expressions has practical applications beyond the classroom. From solving real-world problems involving projectile motion to optimizing engineering designs, the principles of factoring play a crucial role in various fields. By developing a strong foundation in factoring, you're not just mastering an algebraic technique; you're equipping yourself with a powerful tool for problem-solving in diverse contexts.
In conclusion, our exploration of factoring 2x² - 11x - 6 has been a rewarding exercise in algebraic manipulation and critical thinking. We have not only identified the correct factored form but also gained a deeper appreciation for the underlying principles and strategies involved in factoring quadratic expressions. Remember to continue practicing and applying these techniques, and you'll be well-equipped to tackle any factoring challenge that comes your way. The correct answer is indeed A. (2x + 1)(x - 6). This final answer underscores the accuracy of our step-by-step process and the importance of careful evaluation in mathematical problem-solving.
