Factoring Quadratics A Step-by-Step Guide To Solve 6x² - 13x - 5
Introduction: Mastering Quadratic Factorization
In the realm of algebra, factoring quadratic expressions is a fundamental skill, acting as a cornerstone for solving equations, simplifying expressions, and tackling advanced mathematical concepts. One common type of quadratic expression is in the form of ax² + bx + c, where a, b, and c are constants. The question at hand asks us to determine the completely factored form of the quadratic expression 6x² - 13x - 5. This means we need to break down the expression into two binomials, which, when multiplied together, will result in the original quadratic. Let's delve into the step-by-step process of factoring this expression, exploring different methods and highlighting key strategies to ensure a solid understanding of the concept. Understanding quadratic expressions like 6x² - 13x - 5 is essential for various mathematical applications, from solving equations to simplifying complex algebraic problems. Factoring is the process of breaking down such expressions into simpler components, usually binomials, which, when multiplied together, give the original expression. This skill is not just a mathematical exercise; it's a tool that unlocks solutions in many real-world scenarios, such as calculating areas, optimizing designs, and predicting outcomes in various fields. To factor 6x² - 13x - 5, we aim to find two binomials of the form (Ax + B) and (Cx + D) such that their product equals 6x² - 13x - 5. The challenge lies in determining the correct values for A, B, C, and D. This process often involves trial and error, combined with a strategic approach to narrow down the possibilities. Different methods, such as the AC method or trial and error, can be used to find these factors. Each method has its advantages, and choosing the right one can simplify the factoring process. The key is to understand the underlying principles and apply them systematically. In the following sections, we will explore these methods in detail, providing a clear pathway to factoring 6x² - 13x - 5 and similar quadratic expressions.
Method 1: The AC Method - A Systematic Approach to Factoring
The AC method, a widely used technique for factoring quadratic expressions, offers a systematic approach to breaking down the expression. This method is particularly effective when dealing with quadratics where the leading coefficient (the 'a' in ax² + bx + c) is not 1. In our case, the quadratic is 6x² - 13x - 5, so a = 6, b = -13, and c = -5. The first step in the AC method involves multiplying the leading coefficient (a) by the constant term (c). In our example, this means multiplying 6 by -5, which gives us -30. This product, -30, is the key to unlocking the factorization. Next, we need to find two numbers that multiply to -30 and add up to the coefficient of the x term, which is -13. This is where careful consideration comes into play. We need to think of factor pairs of -30 and see which pair sums to -13. The pairs are: (1, -30), (-1, 30), (2, -15), (-2, 15), (3, -10), and (-3, 10), (5, -6), (-5, 6). Upon examining these pairs, we find that the pair (2, -15) satisfies both conditions: 2 * -15 = -30 and 2 + (-15) = -13. Once we've identified these numbers, we rewrite the middle term (-13x) of the quadratic expression as the sum of two terms using these numbers. So, -13x becomes 2x - 15x. Our expression now looks like this: 6x² + 2x - 15x - 5. This step is crucial as it allows us to factor by grouping. Factoring by grouping involves splitting the expression into two pairs of terms and factoring out the greatest common factor (GCF) from each pair. In the first pair, 6x² + 2x, the GCF is 2x. Factoring this out gives us 2x(3x + 1). In the second pair, -15x - 5, the GCF is -5. Factoring this out gives us -5(3x + 1). Notice that both pairs now have a common binomial factor, which is (3x + 1). This is a good sign that we're on the right track. The final step is to factor out the common binomial factor (3x + 1) from the entire expression. This gives us (3x + 1)(2x - 5). Therefore, the completely factored form of 6x² - 13x - 5 using the AC method is (2x - 5)(3x + 1). This method, while seemingly complex at first, provides a structured way to factor quadratic expressions, especially those with a leading coefficient other than 1. The key is to systematically apply each step, from finding the correct numbers to factoring by grouping, to arrive at the final factored form. The AC method is a powerful technique for factoring quadratic expressions, particularly when the leading coefficient is not 1. This systematic approach involves several steps, each crucial to arriving at the correct factored form.
Method 2: Trial and Error - An Intuitive Approach to Factoring
The trial and error method offers a more intuitive approach to factoring quadratic expressions. While it may seem less structured than the AC method, it can be quite efficient with practice and a good understanding of the underlying principles of factoring. This method involves making educated guesses about the binomial factors and then checking if their product matches the original quadratic expression. For the quadratic expression 6x² - 13x - 5, we need to find two binomials of the form (Ax + B)(Cx + D) such that when multiplied, they result in the original quadratic. The first step is to consider the factors of the leading coefficient (6) and the constant term (-5). The factors of 6 are 1 and 6, or 2 and 3. The factors of -5 are 1 and -5, or -1 and 5. We can start by trying different combinations of these factors to form our binomials. For example, we might try (2x + 1)(3x - 5) or (2x - 5)(3x + 1). The key is to strategically choose combinations that are likely to produce the correct middle term (-13x) when the binomials are multiplied. To check if a combination is correct, we multiply the binomials using the FOIL method (First, Outer, Inner, Last). For the combination (2x + 1)(3x - 5), the multiplication would be: First: 2x * 3x = 6x² Outer: 2x * -5 = -10x Inner: 1 * 3x = 3x Last: 1 * -5 = -5 Combining these terms, we get 6x² - 10x + 3x - 5, which simplifies to 6x² - 7x - 5. This does not match our original quadratic expression, so this combination is incorrect. Let's try another combination: (2x - 5)(3x + 1). Multiplying these binomials using the FOIL method: First: 2x * 3x = 6x² Outer: 2x * 1 = 2x Inner: -5 * 3x = -15x Last: -5 * 1 = -5 Combining these terms, we get 6x² + 2x - 15x - 5, which simplifies to 6x² - 13x - 5. This matches our original quadratic expression, so the factored form is (2x - 5)(3x + 1). The trial and error method can be faster than the AC method once you become more comfortable with recognizing patterns and making educated guesses. However, it may require more attempts if the factors are not immediately obvious. The key is to be systematic in your approach, keep track of the combinations you've tried, and learn from your mistakes. With practice, you'll develop a better intuition for which combinations are most likely to work, making this method a valuable tool in your factoring arsenal. It's an approach that encourages a deeper understanding of how the coefficients in the binomials relate to the coefficients in the original quadratic expression.
Step-by-Step Solution: Factoring 6x² - 13x - 5
Let's consolidate our understanding by providing a step-by-step solution to factoring the quadratic expression 6x² - 13x - 5. This will serve as a clear guide, combining the principles of both the AC method and the trial and error method. By understanding the solution, you can build confidence in tackling similar problems.
Step 1: Identify the coefficients. In the quadratic expression 6x² - 13x - 5, the coefficients are: a = 6 (the coefficient of x²), b = -13 (the coefficient of x), and c = -5 (the constant term). Identifying these coefficients is the foundation for both the AC method and the trial and error method.
Step 2: Use the AC method to find the key numbers. Multiply a and c: 6 * -5 = -30. We need to find two numbers that multiply to -30 and add up to b, which is -13. As we determined earlier, the numbers are 2 and -15. These numbers are crucial for rewriting the middle term.
Step 3: Rewrite the middle term. Replace -13x with 2x - 15x. The expression becomes: 6x² + 2x - 15x - 5. This step is essential for factoring by grouping.
Step 4: Factor by grouping. Split the expression into two pairs: (6x² + 2x) and (-15x - 5). Factor out the GCF from each pair: From (6x² + 2x), the GCF is 2x, so we get 2x(3x + 1). From (-15x - 5), the GCF is -5, so we get -5(3x + 1). Notice that both pairs have a common binomial factor, (3x + 1).
Step 5: Factor out the common binomial. Factor out (3x + 1) from the entire expression: (3x + 1)(2x - 5). This is the completely factored form of the quadratic expression.
Step 6: Verify the result (Optional). Multiply the binomials to ensure they equal the original expression: (3x + 1)(2x - 5) = 6x² - 15x + 2x - 5 = 6x² - 13x - 5. This confirms that our factored form is correct.
Alternatively, we could have used the trial and error method by systematically trying different combinations of factors of 6 and -5 until we found the correct combination. This involves testing various binomial pairs until the correct middle term is achieved. The key is to be organized and methodical in your attempts. By following these steps, you can confidently factor quadratic expressions like 6x² - 13x - 5. The combination of the AC method and the trial and error approach provides a robust strategy for solving a wide range of factoring problems. The step-by-step approach not only helps in solving the problem at hand but also builds a strong foundation for tackling more complex algebraic challenges.
Identifying the Correct Answer: (2x - 5)(3x + 1)
After meticulously applying the AC method and verifying our results, we've confidently arrived at the completely factored form of 6x² - 13x - 5, which is (2x - 5)(3x + 1). This outcome aligns perfectly with option A in the given choices. To further solidify our understanding, let's take a moment to consider why the other options are incorrect. This process of elimination not only reinforces the correct solution but also deepens our comprehension of the factoring process. Option B, (2x + 5)(3x - 1), when multiplied out, yields 6x² + 13x - 5. Notice the sign difference in the middle term (+13x instead of -13x), indicating that this is not the correct factorization. Option C, (2x - 1)(3x - 5), results in 6x² - 13x + 5. The constant term here is +5, while our original expression has a constant term of -5, making this option incorrect. Lastly, option D, (2x + 1)(3x + 5), expands to 6x² + 13x + 5. Again, the sign of the middle term and the constant term do not match our original expression, disqualifying this option. By systematically eliminating the incorrect options, we gain a clearer appreciation for the precision required in factoring. The correct factorization, (2x - 5)(3x + 1), accurately captures the relationship between the coefficients and the constant term in the original quadratic expression. The ability to identify the correct answer is not just about finding the solution; it's about understanding why the other options are incorrect. This critical thinking skill is essential for success in mathematics and beyond. By comparing the factored forms with the original expression, we reinforce our understanding of the factoring process and build confidence in our ability to solve similar problems. The process of elimination, coupled with a thorough understanding of factoring techniques, ensures that we not only arrive at the correct answer but also develop a deeper appreciation for the underlying mathematical principles.
Conclusion: Mastering Factoring for Mathematical Success
In conclusion, the completely factored form of the quadratic expression 6x² - 13x - 5 is indeed (2x - 5)(3x + 1), as demonstrated through both the AC method and the trial and error method. This exercise has provided a comprehensive exploration of factoring quadratic expressions, highlighting the importance of systematic approaches and attention to detail. Mastering factoring is not merely a mathematical skill; it's a gateway to unlocking more advanced concepts in algebra and beyond. The ability to factor quadratic expressions efficiently and accurately is essential for solving equations, simplifying expressions, and tackling a wide range of mathematical problems. Whether it's solving quadratic equations, graphing parabolas, or simplifying rational expressions, factoring plays a pivotal role. By understanding the underlying principles of factoring, you empower yourself to approach these challenges with confidence and precision. Moreover, the problem-solving skills honed through factoring extend far beyond the realm of mathematics. The ability to break down complex problems into simpler components, identify patterns, and systematically work towards a solution are valuable assets in any field. From engineering and computer science to finance and economics, the analytical thinking skills developed through factoring are highly sought after. As you continue your mathematical journey, remember that practice is key. The more you factor quadratic expressions, the more intuitive the process becomes. Experiment with different methods, challenge yourself with increasingly complex problems, and don't be afraid to make mistakes – they are valuable learning opportunities. The mastery of factoring is a testament to your mathematical prowess and a foundation for future success. It's a skill that will serve you well in your academic pursuits, your professional endeavors, and your everyday problem-solving. By embracing the challenges of factoring and celebrating the triumphs, you cultivate a lifelong appreciation for the beauty and power of mathematics. The mastery of factoring is a crucial step towards mathematical proficiency, opening doors to a deeper understanding of algebraic concepts and fostering problem-solving skills that extend beyond the classroom.
