Factoring $7x^2 + 13x - 2$: A Step-by-Step Guide

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Hey guys! Today, we're diving into factoring a quadratic expression. Specifically, we're going to break down how to factor $7x^2 + 13x - 2$. Factoring quadratics might seem tricky at first, but with a systematic approach, it becomes much more manageable. This guide will walk you through each step, so you’ll be factoring like a pro in no time! Let's get started!

Understanding Quadratic Expressions

Before we jump into factoring, let's quickly recap what a quadratic expression is. A quadratic expression is a polynomial of degree two, generally written in the form $ax^2 + bx + c$, where a, b, and c are constants, and a is not zero. In our case, we have $7x^2 + 13x - 2$, where $a = 7$, $b = 13$, and $c = -2$. Mastering the art of factoring these expressions is super important in algebra and calculus, as it helps simplify equations and solve for unknowns.

Factoring is essentially the reverse process of expanding. When we expand, we multiply out terms, like going from $(x + 2)(x + 3)$ to $x^2 + 5x + 6$. Factoring is the opposite: we start with $x^2 + 5x + 6$ and want to find $(x + 2)(x + 3)$. There are several techniques for factoring, and the best one to use often depends on the specific quadratic expression you're dealing with. For our expression, $7x^2 + 13x - 2$, we'll use a method that's commonly referred to as the "AC method" or factoring by grouping. This method is particularly useful when the coefficient of $x^2$ (our a term) is not 1.

Understanding this foundational concept is crucial. Without it, attempting to factor can feel like navigating a maze blindfolded. Think of factoring as a puzzle where you’re trying to find the pieces that multiply back together to give you the original expression. So, with our expression $7x^2 + 13x - 2$ in mind, let's move on to the step-by-step process of actually factoring it.

Step 1: Identify a, b, and c

The first step in factoring any quadratic expression is to correctly identify the coefficients a, b, and c. This is super straightforward. In our expression, $7x^2 + 13x - 2$, we have:

• a = 7 (the coefficient of $x^2$)
• b = 13 (the coefficient of $x$)
• c = -2 (the constant term)

It's vital to get these right because they form the basis for the next steps. A small mistake here can throw off the entire factoring process. Once you have a, b, and c clearly identified, you're ready to move on to the crucial step of finding two numbers that meet specific criteria. This is where the magic really begins to happen in the factoring process, so let’s make sure we’re solid on this foundational piece before we proceed.

Think of a, b, and c as the key ingredients in a recipe. If you measure them incorrectly, the final dish won’t taste right. Similarly, if you misidentify these coefficients, your factored expression won’t be correct. So, double-check, triple-check, and make sure you’ve got them right! Now, let's move on to the next step and see how we use these coefficients to unlock the factored form of our quadratic expression.

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

This is the heart of the AC method. We need to find two numbers that satisfy two conditions:

1. Their product must equal ac (which is 7 * -2 = -14 in our case).
2. Their sum must equal b (which is 13 in our case).

Finding these numbers might take a little bit of trial and error, but there's a systematic way to approach it. Start by listing the factor pairs of ac (in this case, -14). We have:

• 1 and -14
• -1 and 14
• 2 and -7
• -2 and 7

Now, we need to check which of these pairs adds up to b (13). Looking at the list, we can see that -1 and 14 fit the bill. -1 multiplied by 14 is -14 (our ac), and -1 plus 14 is 13 (our b). Bingo! We've found our two numbers. This step is crucial because these two numbers will allow us to rewrite the middle term of our quadratic expression, which is the next step in the factoring process.

The ability to quickly identify these numbers is a skill that improves with practice. Don’t get discouraged if you don’t find them immediately. Sometimes it helps to write out all the factor pairs, as we did, to visualize your options. Once you've nailed this step, the rest of the factoring process becomes much smoother. So, with our magic numbers of -1 and 14 in hand, let’s move on to the next step and see how we use them to rewrite our expression.

Step 3: Rewrite the Middle Term

Now that we've found our two magic numbers (-1 and 14), we're going to use them to rewrite the middle term of our quadratic expression. Our original expression is $7x^2 + 13x - 2$. We're going to replace the $13x$ term with $-1x + 14x$. This might seem a bit odd at first, but it's a crucial step in the AC method because it sets us up for factoring by grouping.

So, we rewrite the expression as:

$7x^2 - 1x + 14x - 2$

Notice that we haven't actually changed the value of the expression; we've just rewritten it in a way that makes it easier to factor. Think of it like rearranging the furniture in a room – the room is still the same size, but the layout is different. This step is all about setting ourselves up for success in the next stage of factoring, which is grouping. By breaking down the middle term, we create pairs of terms that share common factors, making the factoring process much more manageable.

This rewriting of the middle term is the linchpin of the AC method. It's the bridge between the initial quadratic expression and its factored form. Without it, we'd be stuck with an expression that's difficult to factor directly. So, make sure you're comfortable with this step before moving on. Once we've rewritten the middle term, we can move on to the next phase: factoring by grouping. This is where we'll actually start pulling out common factors and simplifying the expression.

Step 4: Factor by Grouping

With our rewritten expression, $7x^2 - 1x + 14x - 2$, we're ready to factor by grouping. This involves grouping the first two terms and the last two terms together and then factoring out the greatest common factor (GCF) from each group.

Let's group the terms:

$(7x^2 - 1x) + (14x - 2)$

Now, we'll factor out the GCF from each group.

From the first group, $(7x^2 - 1x)$, the GCF is x. Factoring out x, we get:

$x(7x - 1)$

From the second group, $(14x - 2)$, the GCF is 2. Factoring out 2, we get:

$2(7x - 1)$

So, our expression now looks like:

$x(7x - 1) + 2(7x - 1)$

Notice something cool? Both terms have a common factor of $(7x - 1)$. This is exactly what we want! It means we're on the right track. Factoring by grouping is all about creating this common binomial factor. Once you see that common factor emerge, you know you're close to the finish line. This step is where the hard work of the previous steps really pays off, as we can now see the structure of the factored expression taking shape.

This technique of grouping terms and factoring out common factors is a powerful tool in algebra. It allows us to break down complex expressions into simpler components, making them easier to work with. So, let’s take that common factor and move on to the final step of factoring our quadratic expression.

Step 5: Factor Out the Common Binomial

We've arrived at the final step! Looking at our expression, $x(7x - 1) + 2(7x - 1)$, we can see that $(7x - 1)$ is a common binomial factor. We can factor this out just like we factored out the GCF in the previous step.

Factoring out $(7x - 1)$, we get:

$(7x - 1)(x + 2)$

And there you have it! We've successfully factored the quadratic expression $7x^2 + 13x - 2$ into $(7x - 1)(x + 2)$. This is the factored form of our original expression. We’ve taken a quadratic trinomial and broken it down into the product of two binomials.

This final step is like putting the last piece in a puzzle. It's the culmination of all the previous steps, and it gives us the complete picture. Factoring out the common binomial brings everything together and reveals the factors that make up the original quadratic expression. So, congratulations, you’ve made it through the factoring process! But, as with any mathematical process, it’s always a good idea to check our work.

Checking Our Work

To make sure we've factored correctly, we can expand our factored expression $(7x - 1)(x + 2)$ and see if we get back our original expression, $7x^2 + 13x - 2$.

Expanding $(7x - 1)(x + 2)$ using the distributive property (or the FOIL method), we get:

$7x(x) + 7x(2) - 1(x) - 1(2)$

Which simplifies to:

$7x^2 + 14x - x - 2$

Combining like terms, we get:

$7x^2 + 13x - 2$

This matches our original expression! So, we know our factoring is correct. This check is a crucial step in the factoring process. It's like a safety net that ensures we haven't made any mistakes along the way. By expanding our factored expression, we can confirm that it's equivalent to the original quadratic expression, giving us confidence in our solution.

Checking our work isn't just about finding mistakes; it's also a great way to reinforce our understanding of the factoring process. By going back and forth between the factored form and the expanded form, we solidify our grasp of the relationship between the two. So, always take the time to check your work, and you'll become a factoring master in no time!

Conclusion

Factoring the quadratic expression $7x^2 + 13x - 2$ involves a few key steps: identifying a, b, and c, finding two numbers that multiply to ac and add to b, rewriting the middle term, factoring by grouping, and finally, factoring out the common binomial. By following these steps, we successfully factored the expression into $(7x - 1)(x + 2)$. And, of course, we checked our work to make sure we got it right!

Factoring quadratics might seem challenging initially, but with practice and a clear understanding of the steps involved, you'll become more comfortable and confident. Remember, the key is to break down the problem into manageable steps and to check your work along the way. Keep practicing, and you'll be factoring even the trickiest quadratics in no time!

So, there you have it, guys! We've walked through the process of factoring $7x^2 + 13x - 2$ step by step. I hope this guide has been helpful and has made factoring a little less intimidating. Keep practicing, and you'll be a factoring whiz in no time! Happy factoring!