Factoring: Solve 2x^2 + 13x = -6 Quadratic Equation

Hey guys! Let's dive into the world of quadratic equations and tackle a common problem: solving by factoring. We’re going to specifically look at the equation 2x² + 13x = -6. This might seem daunting at first, but don't worry! We'll break it down step by step, so you’ll be solving these like a pro in no time. So, buckle up, grab your pencils, and let's get started!

Why Factoring Matters

Before we jump into the nitty-gritty, let’s quickly talk about why factoring is such a crucial skill in algebra. Factoring is like reverse distribution. It's a way of breaking down a complex expression into simpler parts. When it comes to quadratic equations, factoring helps us find the roots, or the solutions, of the equation. These roots are the values of 'x' that make the equation true. Understanding this method not only helps in solving equations but also builds a strong foundation for more advanced math topics.

Factoring, at its core, is about rewriting a quadratic expression as a product of two binomials. Think of it like this: we're trying to undo the multiplication that created the quadratic. This skill is incredibly useful because once we have the equation in factored form, we can easily find the solutions by setting each factor equal to zero. This is based on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This property is the key that unlocks the solutions to our quadratic equations when we factor them. Mastering factoring opens doors to solving a wide array of problems in algebra and beyond, making it an indispensable tool in your mathematical toolkit.

Moreover, factoring isn't just a standalone technique; it's a gateway to understanding other methods of solving quadratic equations, such as completing the square and using the quadratic formula. Each method has its strengths and weaknesses, and the ability to recognize when factoring is the most efficient approach is a valuable skill. For instance, if a quadratic equation can be easily factored, it's often quicker to solve it that way than to resort to the quadratic formula. Factoring also enhances your understanding of polynomial manipulation, which is crucial for higher-level math courses. It's a fundamental skill that underpins many algebraic concepts, making it well worth the effort to master.

Step 1: Setting the Stage – Get the Equation into Standard Form

Okay, so the first thing we need to do with our equation 2x² + 13x = -6 is to get it into the standard quadratic form. Remember that the standard form looks like this: ax² + bx + c = 0. This is crucial because factoring techniques are designed to work with equations in this format. It's like making sure all the ingredients are prepped before you start cooking; you need the equation in the right form before you can start factoring.

In our case, we have 2x² + 13x = -6. Notice that the right side isn't zero yet. No worries, we can easily fix this! To get the equation into standard form, we need to move that '-6' over to the left side. How do we do that? Simple! We add 6 to both sides of the equation. This keeps the equation balanced and gets us closer to our goal.

Adding 6 to both sides gives us: 2x² + 13x + 6 = 0. Ta-da! Now our equation is in the beautiful standard form we need. See, that wasn't so hard, was it? Getting the equation into this form is the essential first step because it allows us to clearly identify the coefficients 'a', 'b', and 'c', which are necessary for the factoring process. In our equation, 'a' is 2, 'b' is 13, and 'c' is 6. Keep these values in mind as we move on to the next step. This meticulous preparation is what sets the stage for a smooth and successful factoring journey.

Step 2: The Factoring Puzzle – Finding the Right Numbers

Now comes the fun part: actually factoring the quadratic expression! We've got 2x² + 13x + 6 = 0 in standard form, and we need to break it down into two binomials. This is where the factoring puzzle begins. The goal is to find two binomials that, when multiplied together, give us our original quadratic expression. Think of it as reverse engineering – we're trying to figure out what two expressions were multiplied to get our current expression.

There are different methods for factoring, but one common approach is to look for two numbers that multiply to a*c and add up to b. Remember our 'a', 'b', and 'c' from the previous step? In our equation, a = 2, b = 13, and c = 6. So, we need to find two numbers that multiply to 2 * 6 = 12 and add up to 13. This might sound like a daunting task, but let's systematically think through the factors of 12.

The factors of 12 are: 1 and 12, 2 and 6, 3 and 4. Which pair adds up to 13? Bingo! It's 1 and 12. This is a crucial step, so take your time and practice this number-finding game. Once you've identified the correct pair, you're one step closer to cracking the factoring code. These numbers are the key to rewriting the middle term (13x) in a way that allows us to factor by grouping. This technique involves splitting the middle term into two terms using the numbers we just found, and then factoring out common factors from pairs of terms. It's like dissecting the quadratic expression to reveal its hidden structure.

Step 3: Rewrite and Regroup

With our magic numbers 1 and 12 in hand, we can rewrite our middle term, 13x. Instead of writing 13x, we're going to express it as 1x + 12x. This might seem like we're just making things more complicated, but trust me, it's a clever trick that will lead us to the factored form. So, our equation 2x² + 13x + 6 = 0 now becomes 2x² + 1x + 12x + 6 = 0. See how we've just split the 13x into two terms using our numbers?

Now comes the regrouping part. We're going to group the first two terms together and the last two terms together. This is like forming two mini-expressions within our larger expression. We group them using parentheses: (2x² + 1x) + (12x + 6) = 0. The parentheses act like little containers, keeping the terms together as we prepare to factor out common factors. This step is essential because it sets the stage for factoring by grouping, a technique that allows us to break down the quadratic expression into smaller, more manageable pieces. Think of it as organizing your tools before starting a project; it makes the whole process smoother and more efficient. By regrouping, we create opportunities to identify common factors within each pair of terms, which is the key to the next step.

Step 4: Factor by Grouping

Alright, we've regrouped our equation into (2x² + 1x) + (12x + 6) = 0. Now it's time to factor out the greatest common factor (GCF) from each group. Remember, the GCF is the largest term that divides evenly into all terms in the group. Factoring out the GCF is like peeling back the layers of the expression to reveal its underlying structure. It's a crucial step in breaking down the quadratic into its factored form.

Looking at the first group, (2x² + 1x), what's the GCF? Well, both terms have an 'x' in them, so we can factor out an 'x'. This gives us x(2x + 1). See how we've taken the common factor 'x' out of both terms? Now, let's move on to the second group, (12x + 6). What's the GCF here? Both 12x and 6 are divisible by 6, so we can factor out a 6. This gives us 6(2x + 1). Notice anything interesting? Both groups now have a common binomial factor: (2x + 1). This is a sign that we're on the right track! The appearance of the same binomial factor in both groups is a telltale sign that factoring by grouping is working effectively. It's like finding a matching puzzle piece that fits perfectly into both sections. This common binomial factor is the key to completing the factoring process, as it allows us to combine the terms outside the parentheses into another factor.

Step 5: The Final Factor – Putting It All Together

We've done the hard work of factoring out the GCFs, and we're now at the final factoring stage. Our equation looks like this: x(2x + 1) + 6(2x + 1) = 0. Notice that both terms have a common binomial factor of (2x + 1). This is fantastic because it means we can factor this binomial out of the entire expression. It's like pulling out the common thread that runs through the entire equation.

To do this, we treat the binomial (2x + 1) as a single term and factor it out. This leaves us with (2x + 1)(x + 6) = 0. Boom! We've factored the quadratic equation! This is the moment of triumph where we see the original quadratic expression transformed into a product of two binomials. This factored form is incredibly powerful because it allows us to easily find the solutions to the equation. It's like unlocking the secret code that reveals the roots of the quadratic.

The factored form (2x + 1)(x + 6) = 0 represents the original quadratic expression rewritten as a product of two binomial factors. This is the culmination of our factoring efforts, and it sets the stage for the final step: finding the values of 'x' that make the equation true. This factored form is not just an algebraic manipulation; it's a representation of the underlying structure of the quadratic equation, showing how it can be broken down into simpler, linear components.

Step 6: Solving for x – The Zero Product Property

We've got our factored equation: (2x + 1)(x + 6) = 0. Now it's time to find the values of 'x' that make this equation true. This is where the zero product property comes into play. Remember, this property states that if the product of two factors is zero, then at least one of the factors must be zero. This is the golden rule that allows us to solve for 'x' once we have the equation in factored form. It's like having a magic key that unlocks the solutions to our quadratic puzzle.

So, we set each factor equal to zero and solve for 'x'. First, let's take the factor (2x + 1) and set it equal to zero: 2x + 1 = 0. To solve for 'x', we subtract 1 from both sides, giving us 2x = -1. Then, we divide both sides by 2, which gives us our first solution: x = -1/2. Now, let's do the same for the second factor, (x + 6). We set it equal to zero: x + 6 = 0. To solve for 'x', we subtract 6 from both sides, giving us our second solution: x = -6. And there you have it! We've found the two values of 'x' that make the equation 2x² + 13x = -6 true.

These solutions, x = -1/2 and x = -6, are the roots of the quadratic equation. They are the points where the parabola represented by the equation intersects the x-axis. Finding these solutions is the ultimate goal of solving a quadratic equation, and the zero product property is the tool that makes it possible. This property transforms the problem of solving a quadratic equation into two simpler problems of solving linear equations, making the whole process much more manageable.

Conclusion: Factoring Success!

Woohoo! We did it! We successfully solved the quadratic equation 2x² + 13x = -6 by factoring. We took it step by step, from getting the equation into standard form to using the zero product property to find our solutions. Remember, the key steps are: getting the equation in standard form, finding the right numbers that multiply to a*c and add up to b, rewriting and regrouping, factoring by grouping, and finally, using the zero product property to solve for x.

Factoring might seem tricky at first, but with practice, you'll become a factoring master! Keep practicing, and you'll be able to tackle any quadratic equation that comes your way. And remember, math is not just about getting the right answer; it's about the journey of problem-solving and building your understanding. So, keep exploring, keep questioning, and keep learning. You've got this!

If you found this guide helpful, share it with your friends and classmates, and let's conquer the world of quadratic equations together! Happy factoring!