Factoring Quadratics A Step By Step Guide To $6x^2 - 11x - 10$

Hey everyone! Today, let's dive into factoring the quadratic expression $6x^2 - 11x - 10$. Factoring quadratics is a crucial skill in algebra, and it's super useful for solving equations and simplifying expressions. We'll break down the steps and make sure you understand exactly how to tackle this type of problem. So, let's get started!

Understanding the Problem

Before we jump into the solution, let's quickly understand what we're dealing with. We have a quadratic expression in the form of $ax^2 + bx + c$, where:

• $a = 6$
• $b = -11$
• $c = -10$

Our goal is to rewrite this expression as a product of two binomials, something like $(px + q)(rx + s)$. There are several methods to do this, but we'll focus on the factoring by grouping method, which is both effective and widely used.

Why Factoring by Grouping?

The factoring by grouping method is particularly handy when the leading coefficient (in our case, $a = 6$) is not 1. It helps break down the problem into smaller, more manageable steps. Plus, once you get the hang of it, it becomes a go-to technique for factoring complex quadratics. We will explore each step meticulously to ensure clarity and mastery.

Step-by-Step Solution

Step 1: Multiply $a$ and $c$

First, we multiply the leading coefficient $a$ by the constant term $c$:

$a \times c = 6 \times (-10) = -60$

This product, $-60$, is a key number that we'll use in the next step. It acts as the target product we need to achieve when we break down the middle term. This initial multiplication sets the stage for finding the right factors that will allow us to rewrite and group the quadratic expression effectively. The negative sign is crucial, as it indicates that we need to find two factors with opposite signs.

Step 2: Find Two Numbers That Multiply to $ac$ and Add Up to $b$

Now, we need to find two numbers that:

• Multiply to $-60$
• Add up to $b = -11$

This is where a little bit of number sense comes in handy. We're looking for two numbers with a significant difference because their product is negative, and their sum is also negative. Let's list some factor pairs of $60$:

• $1$ and $60$
• $2$ and $30$
• $3$ and $20$
• $4$ and $15$
• $5$ and $12$
• $6$ and $10$

Among these pairs, $4$ and $15$ seem promising because their difference can be close to $11$. To get a sum of $-11$, we need the numbers $-15$ and $4$:

• $(-15) \times 4 = -60$
• $(-15) + 4 = -11$

So, we've found our numbers! This step is pivotal because these two numbers will help us rewrite the middle term of the quadratic expression, which is the foundation of the grouping method. Identifying the correct pair of numbers is often the most challenging part, but with practice, it becomes more intuitive.

Step 3: Rewrite the Middle Term

We'll rewrite the middle term, $-11x$, using the two numbers we just found, $-15$ and $4$:

$6x^2 - 11x - 10 = 6x^2 - 15x + 4x - 10$

Notice that we've replaced $-11x$ with $-15x + 4x$. This doesn't change the expression's value, but it sets us up perfectly for the next step: grouping. Rewriting the middle term in this way is the cornerstone of factoring by grouping, as it allows us to break the expression down into pairs that share common factors. This strategic rewrite is what transforms a trinomial into a four-term expression that can be easily factored.

Step 4: Factor by Grouping

Now, we group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

$(6x^2 - 15x) + (4x - 10)$

From the first group, $6x^2 - 15x$, the GCF is $3x$. Factoring this out, we get:

$3x(2x - 5)$

From the second group, $4x - 10$, the GCF is $2$. Factoring this out, we get:

$2(2x - 5)$

So, our expression now looks like this:

$3x(2x - 5) + 2(2x - 5)$

The key observation here is that both terms have a common factor of $(2x - 5)$. This is exactly what we want, as it allows us to factor this common binomial out of the entire expression. Factoring by grouping is all about identifying and extracting these common binomial factors, which leads us to the final factorization.

Step 5: Factor Out the Common Binomial

We can now factor out the common binomial $(2x - 5)$ from the entire expression:

$(2x - 5)(3x + 2)$

And that's it! We've successfully factored the quadratic expression. This final step is where all the previous work comes together, resulting in the factored form of the original expression. It’s a satisfying moment when you can see how the individual steps have combined to give you the solution.

Final Answer

So, the completely factored form of $6x^2 - 11x - 10$ is:

$(3x + 2)(2x - 5)$

Therefore, the correct answer is C. $(3x + 2)(2x - 5)$.

Checking Our Work

It's always a good idea to check our work by expanding the factored form to make sure we get back the original expression. Let's multiply $(3x + 2)(2x - 5)$ using the FOIL method:

• First: $(3x)(2x) = 6x^2$
• Outer: $(3x)(-5) = -15x$
• Inner: $(2)(2x) = 4x$
• Last: $(2)(-5) = -10$

Combining these, we get:

$6x^2 - 15x + 4x - 10 = 6x^2 - 11x - 10$

This matches our original expression, so we know we've factored it correctly. Checking your work is a crucial habit in algebra, as it helps you catch any mistakes and ensures that you’re confident in your solution. This simple expansion can save you from errors and reinforce your understanding of factoring.

Common Mistakes to Avoid

When factoring quadratic expressions, it’s easy to make a few common mistakes. Here are a few to watch out for:

• Sign Errors: Pay close attention to the signs of the numbers you're using. A simple sign mistake can throw off the entire factoring process. Always double-check that the signs match up correctly when you're rewriting the middle term and factoring out common factors.
• Incorrect GCF: Make sure you're factoring out the greatest common factor from each group. If you don't factor out the greatest common factor, you might end up with an expression that's not fully factored.
• Forgetting to Check: Always, always, always check your answer by expanding the factored form. This is the easiest way to catch mistakes and ensure that you've factored the expression correctly. Checking your work is like having a safety net—it can save you from submitting an incorrect answer.
• Mixing Up the Order: When rewriting the middle term, ensure that you keep the order consistent. For example, if you rewrite $-11x$ as $-15x + 4x$, make sure you group the terms correctly in the next step. Mixing up the order can lead to incorrect grouping and a wrong final answer.

By being mindful of these common pitfalls, you can improve your accuracy and confidence in factoring quadratic expressions.

Practice Problems

To really nail this skill, practice is key! Here are a few more quadratic expressions you can try factoring:

1. $2x^2 + 7x + 3$
2. $3x^2 - 8x + 4$
3. $4x^2 + 12x + 5$

Work through these problems using the same steps we covered, and don't forget to check your answers. The more you practice, the more comfortable and proficient you'll become with factoring. Each problem you solve helps solidify your understanding and builds your problem-solving skills. Happy factoring, guys!

Conclusion

Factoring quadratic expressions like $6x^2 - 11x - 10$ might seem tricky at first, but with a step-by-step approach, it becomes much more manageable. Remember to multiply $a$ and $c$, find the right factors, rewrite the middle term, group the terms, factor out the common factors, and always check your work. With practice, you'll become a factoring pro! This skill is incredibly useful in algebra and beyond, so mastering it is well worth the effort. Keep practicing, and you'll be able to tackle any quadratic that comes your way. Good luck, and keep up the great work!