Finding Factors: A Step-by-Step Guide

Hey math enthusiasts! Ever get stuck on a factoring problem and wish there was a clear path to the solution? Well, you're in the right place! Today, we're diving deep into the world of factoring, specifically looking at the quadratic expression $10x^2 - 11x - 6$. Our mission? To identify which of the provided options – A. $5x - 3$, B. $5x + 2$, C. $2x + 1$, or D. None of the above – is a factor of this expression. Don't worry if the term "factor" sounds a bit intimidating; we'll break it down into easy-to-understand steps. We'll explore different methods to find the factors and then choose the correct option from the given choices. Let's get started and unravel the mysteries of factoring together. This guide will walk you through the process, making sure you grasp every detail. By the end, you'll be confident in tackling similar problems, turning what seemed complex into something straightforward. So, grab your pencils, and let's start the journey!

Understanding Factors and Factoring

Let's begin with the basics. What are factors? In mathematics, a factor is a number or an expression that divides another number or expression without any remainder. Think of it like this: if you can multiply two numbers to get a third number, those two numbers are factors of the third number. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12, because each of these numbers divides 12 evenly. Now, when we talk about factoring, we're essentially doing the opposite of multiplying. We're breaking down a mathematical expression into its constituent factors. This is particularly useful when dealing with quadratic equations, which are equations that can be written in the form $ax^2 + bx + c = 0$, where a, b, and c are constants, and $x$ is the variable. Factoring helps us find the values of $x$ that satisfy the equation. In our specific problem, $10x^2 - 11x - 6$, our goal is to find expressions that, when multiplied together, give us the original quadratic expression. This means we're looking for two binomials (expressions with two terms) that multiply to give $10x^2 - 11x - 6$. There are several ways to factor quadratic expressions, and we'll explore one of the most reliable methods: the AC method, in the following sections. Understanding this concept is crucial, and it's the foundation upon which we'll build our solution. It makes the entire process of identifying the correct option much simpler.

The AC Method: A Detailed Approach

The AC method is a powerful technique for factoring quadratic expressions. This method involves several steps that help us break down the quadratic expression into its factors systematically. Let's apply this method to our expression, $10x^2 - 11x - 6$. First, we need to identify the coefficients a, b, and c from our quadratic expression, which is in the standard form $ax^2 + bx + c$. In our case, a = 10, b = -11, and c = -6. The first step in the AC method is to multiply 'a' and 'c'. So, we calculate $a * c = 10 * (-6) = -60$. The next step involves finding two numbers that multiply to give -60 (the result from the previous step) and add up to -11 (the value of 'b'). This step is often the most challenging, as it requires some trial and error. Let's think about the factors of -60: (1, -60), (-1, 60), (2, -30), (-2, 30), (3, -20), (-3, 20), (4, -15), (-4, 15), (5, -12), (-5, 12), (6, -10), (-6, 10). Among these pairs, the numbers 4 and -15 add up to -11 (4 + (-15) = -11). Now, we rewrite the middle term (-11x) using these two numbers. Our expression $10x^2 - 11x - 6$ becomes $10x^2 + 4x - 15x - 6$. Next, we use grouping. We group the first two terms and the last two terms: $(10x^2 + 4x) + (-15x - 6)$. Now, we factor out the greatest common factor (GCF) from each group. For the first group, the GCF is $2x$, and for the second group, the GCF is -3. This gives us $2x(5x + 2) - 3(5x + 2)$. Notice that both terms now have a common factor of $(5x + 2)$. Finally, we factor out $(5x + 2)$, which gives us $(5x + 2)(2x - 3)$. So, the factors of $10x^2 - 11x - 6$ are $(5x + 2)$ and $(2x - 3)$. This detailed breakdown shows how the AC method guides us to the correct factors, providing a clear path to the solution.

Applying the Method to the Options

Now that we have successfully factored the given quadratic expression using the AC method, it's time to evaluate the given options and choose the correct factor. We have determined that the factors of $10x^2 - 11x - 6$ are $(5x + 2)$ and $(2x - 3)$. Now, let's examine the options provided in the question: A. $5x - 3$, B. $5x + 2$, C. $2x + 1$, and D. None of the above. By comparing our findings with the options, we can easily identify the correct one. The factor $(5x + 2)$ is exactly the same as option B. Therefore, the correct answer is option B. Option A, $5x - 3$, is also a factor, but since it is not included in the provided options, we cannot select it. Option C, $2x + 1$, is not a factor of the original quadratic expression. The AC method, as shown, gives us the correct factors through a systematic approach, ensuring we identify the right answer. This method proves that by carefully following the steps, we can confidently determine the factors of a quadratic expression and select the appropriate answer from the given choices. This process highlights how factoring simplifies complex expressions, making them easier to understand and solve.

Step-by-Step Verification

To solidify our understanding and ensure our answer is correct, let's go through the steps one more time and verify our solution. Remember, we started with the quadratic expression $10x^2 - 11x - 6$ and our goal was to find a factor among the options provided. We used the AC method, which involves multiplying the 'a' and 'c' coefficients (10 * -6 = -60) and then finding two numbers that multiply to -60 and add up to -11. We found these numbers to be 4 and -15. We then rewrote the middle term and grouped the terms to factor by grouping. Through this process, we found that the factors of the expression were $(5x + 2)$ and $(2x - 3)$. By comparing our factored form with the answer choices, we matched one of our factors, $(5x + 2)$, to option B. Therefore, the correct answer is B. $5x + 2$. This step-by-step verification confirms the correctness of our solution. By understanding the method and carefully following each step, we have successfully identified the correct factor. This process underscores the value of systematic approaches in solving mathematical problems. It shows how precision and attention to detail can lead to accurate solutions. By repeatedly solving such problems, we sharpen our problem-solving skills and develop a deeper understanding of mathematical concepts.

Conclusion

We have successfully navigated through the process of factoring a quadratic expression and identifying its factors. Starting with the core concept of factors, we moved on to learn the AC method and applied it to our problem, $10x^2 - 11x - 6$. By using the AC method step by step, we factored the expression to find its components and matched one of them to the choices provided. The correct factor was found to be $5x + 2$, corresponding to option B. This exercise demonstrates the power of systematic methods in mathematics and how, with the right approach, complex problems can be simplified. Remember, practice is key. The more you work through factoring problems, the more confident you will become. Keep exploring and applying these methods to similar expressions. This will not only improve your problem-solving skills but also deepen your understanding of algebraic concepts. Factoring might seem challenging initially, but with practice, it becomes a valuable skill in your mathematical toolkit. So, keep at it, and enjoy the journey of learning and discovery! You are now well-equipped to tackle similar problems. Keep practicing and applying these techniques, and you will find yourself becoming more confident and proficient in factoring and related algebraic concepts.

For further practice, you can check out resources on Khan Academy.