Equivalent Expression Of 6x² - 19x - 55 A Step By Step Solution
Hey guys! Today, we're diving deep into a classic algebra problem: finding the expression equivalent to 6x² - 19x - 55. This type of question often appears in math exams, and mastering it is crucial for building a strong foundation in algebra. We will meticulously analyze each option provided, breaking down the process step by step, so you can confidently tackle similar problems in the future. So, let's put on our thinking caps and get started!
Understanding the Problem: Factoring Quadratic Expressions
Before we jump into the options, let's quickly recap what it means to find an equivalent expression. In this case, we're looking for a factored form of the quadratic expression 6x² - 19x - 55. Factoring a quadratic expression means rewriting it as a product of two binomials. Think of it as the reverse of expanding brackets. The goal is to identify two binomials that, when multiplied together, give us the original quadratic expression. This involves a bit of trial and error, strategic thinking, and understanding the relationship between the coefficients in the quadratic expression and the constants in the binomials. This is a fundamental concept in algebra, and mastering it opens doors to solving many other mathematical problems. Let's break down the process and then apply it to our specific question.
Why Factoring Matters
Factoring isn't just a mathematical exercise; it's a powerful tool with real-world applications. Think about engineering, physics, and even economics – quadratic equations pop up everywhere! Being able to factor them quickly and accurately allows us to solve for unknowns, optimize designs, and make predictions. Moreover, factoring is a gateway to understanding more advanced algebraic concepts. It's a foundational skill that builds confidence and sets you up for success in higher-level math courses. Factoring enables us to simplify complex expressions, solve equations, and graph functions more efficiently. It's like having a secret weapon in your mathematical arsenal!
Key Strategies for Factoring
So, how do we approach factoring these tricky quadratics? Here are a few key strategies to keep in mind:
- Look for Common Factors: Always check if there's a common factor you can pull out from all the terms. This simplifies the expression and makes it easier to factor further.
- The AC Method: This is a classic technique that involves multiplying the leading coefficient (A) by the constant term (C) and then finding factors of that product that add up to the middle coefficient (B). It might sound complicated, but it's a systematic approach that works wonders.
- Trial and Error: Sometimes, the best approach is to try different combinations of binomials until you find the right one. This requires a good understanding of how binomial multiplication works.
- The FOIL Method (in Reverse): Remember FOIL (First, Outer, Inner, Last)? We use this when expanding binomials. When factoring, we're essentially working backward, trying to figure out which terms would have produced the original expression.
Applying the Strategies to Our Problem
Now, let's bring these strategies to our problem: 6x² - 19x - 55. First, we see that there's no common factor we can pull out from all three terms. So, let's try the AC method. A is 6, C is -55, and B is -19.
- A * C = 6 * -55 = -330
- We need to find two factors of -330 that add up to -19.
This might seem daunting, but let's break it down. We need one positive and one negative factor since the product is negative. After some careful consideration (and maybe a little trial and error), we find that 11 and -30 fit the bill:
- 11 * -30 = -330
- 11 + (-30) = -19
These are our magic numbers! Now, we use them to rewrite the middle term of our quadratic expression:
6x² - 19x - 55 becomes 6x² + 11x - 30x - 55
See what we did there? We split the -19x into 11x - 30x. This allows us to factor by grouping. Now, we group the first two terms and the last two terms:
(6x² + 11x) + (-30x - 55)
Next, we factor out the greatest common factor from each group:
x(6x + 11) - 5(6x + 11)
Notice that we now have a common binomial factor: (6x + 11). We can factor this out:
(6x + 11)(x - 5)
And there we have it! We've factored the quadratic expression. Now, let's see if this matches any of the options provided.
Analyzing the Options: A Step-by-Step Approach
Now that we've factored the expression ourselves, let's meticulously examine each option to see which one matches our result. This is a critical step, as it allows us to verify our work and ensure we've arrived at the correct answer. We'll use the FOIL method (First, Outer, Inner, Last) to expand each option and compare it to the original expression, 6x² - 19x - 55. This process will not only help us identify the correct answer but also reinforce our understanding of binomial multiplication and factoring.
Option A: (2x - 11)(3x + 5)
Let's expand this using FOIL:
- First: (2x)(3x) = 6x²
- Outer: (2x)(5) = 10x
- Inner: (-11)(3x) = -33x
- Last: (-11)(5) = -55
Now, let's combine the terms: 6x² + 10x - 33x - 55 = 6x² - 23x - 55
This does not match our original expression, 6x² - 19x - 55. So, option A is incorrect. See how expanding and simplifying helps us quickly eliminate incorrect answers? This methodical approach is key to solving these types of problems efficiently.
Option B: (2x + 11)(3x - 5)
Let's expand this one using FOIL:
- First: (2x)(3x) = 6x²
- Outer: (2x)(-5) = -10x
- Inner: (11)(3x) = 33x
- Last: (11)(-5) = -55
Combining the terms: 6x² - 10x + 33x - 55 = 6x² + 23x - 55
Again, this does not match 6x² - 19x - 55. Option B is also incorrect. We're getting closer, though! This process of elimination is a valuable strategy when tackling multiple-choice questions.
Option C: (6x - 11)(x + 5)
Time to expand option C using FOIL:
- First: (6x)(x) = 6x²
- Outer: (6x)(5) = 30x
- Inner: (-11)(x) = -11x
- Last: (-11)(5) = -55
Combining the terms: 6x² + 30x - 11x - 55 = 6x² + 19x - 55
This is close, but the middle term has the wrong sign! We need -19x, not +19x. So, option C is incorrect. This highlights the importance of paying close attention to the signs when expanding and simplifying.
Option D: (6x + 11)(x - 5)
Let's give option D the FOIL treatment:
- First: (6x)(x) = 6x²
- Outer: (6x)(-5) = -30x
- Inner: (11)(x) = 11x
- Last: (11)(-5) = -55
Combining the terms: 6x² - 30x + 11x - 55 = 6x² - 19x - 55
Bingo! This perfectly matches our original expression. Therefore, option D is the correct answer.
The Verdict: Option D is the Winner!
After carefully factoring the expression 6x² - 19x - 55 and meticulously analyzing each option using the FOIL method, we've confidently arrived at the solution: Option D, (6x + 11)(x - 5), is the equivalent expression. We successfully navigated the world of quadratic expressions, factoring, and binomial multiplication. You guys rock!
Key Takeaways and Tips for Success
Before we wrap up, let's recap some key takeaways and tips to help you conquer similar problems in the future:
- Master Factoring Techniques: Understanding techniques like the AC method and factoring by grouping is crucial for tackling quadratic expressions.
- Use FOIL to Verify: Always use the FOIL method to expand the options and verify if they match the original expression.
- Pay Attention to Signs: Be extra careful with signs when expanding and simplifying. A small mistake can lead to an incorrect answer.
- Practice Makes Perfect: The more you practice, the more comfortable you'll become with factoring and expanding expressions. So, keep at it!
- Break It Down: If a problem seems overwhelming, break it down into smaller, manageable steps. This makes the process less daunting and increases your chances of success.
Level Up Your Algebra Skills!
Factoring quadratic expressions is a fundamental skill in algebra, and mastering it will open doors to more advanced concepts. Keep practicing, keep exploring, and never stop learning. Remember, every problem is an opportunity to grow and strengthen your mathematical abilities. You've got this!
I hope this comprehensive guide has helped you understand how to find the equivalent expression for 6x² - 19x - 55. Keep practicing, and you'll become a factoring pro in no time! Good luck with your algebra adventures, and I'll see you in the next math challenge!
