Factor $90x^2+60x-80$: A Step-by-Step Guide

Hey there, math enthusiasts and curious minds! Ever looked at a big, hairy algebraic expression like $90x^2+60x-80$ and felt a little overwhelmed? Don't sweat it, guys! Factoring completely might seem like a daunting task, especially with those larger coefficients, but I promise you, with the right approach, it's totally manageable and even a bit fun. Think of it as breaking down a complex puzzle into smaller, easier-to-solve pieces. In this comprehensive guide, we're going to dive deep into how to factor this specific quadratic expression from start to finish, making sure you understand every single step. We'll break down the process into easy-to-digest chunks, explaining the "why" behind each move, not just the "how." By the end of this article, you'll not only have the solution to $90x^2+60x-80$ but also a solid foundation for tackling similar quadratic factoring challenges with confidence. So, grab your imaginary math superhero cape, and let's get factoring!

Why Factoring Quadratic Expressions Matters

Factoring quadratic expressions is an absolutely fundamental skill in algebra, and trust me, it's way more than just a classroom exercise. When you learn to factor expressions like $90x^2+60x-80$, you're essentially learning to unpack complex mathematical statements, revealing their simpler building blocks. This process is super important for a bunch of reasons. First off, it's your go-to method for solving quadratic equations. Imagine you have an equation where $90x^2+60x-80$ equals zero; factoring allows you to find the values of 'x' that make that statement true, which are often called the roots or zeros of the equation. These roots can tell you where a parabola crosses the x-axis, which is incredibly useful in fields ranging from physics (think projectile motion) to economics (modeling profit functions). Without factoring, solving many of these equations would be much, much harder, often requiring more complex formulas that don't always give you as clear an insight into the structure of the problem. Moreover, factoring helps in simplifying algebraic fractions, making them easier to work with, just like simplifying a fraction like 4/8 to 1/2. It’s also crucial for identifying patterns and relationships between different algebraic expressions, which is a cornerstone of advanced mathematics. When you factor completely, you ensure that you've broken down the expression into its irreducible parts, meaning there's no further simplification possible using integer coefficients. This completeness is key because it gives you the most simplified and insightful form of the expression. So, while our goal here is to specifically factor $90x^2+60x-80$, remember that the skills you develop are broadly applicable and will serve you well throughout your mathematical journey. It's a foundational skill that unlocks deeper understanding and problem-solving power in many areas of math and science, giving you a powerful tool in your algebraic arsenal. Let's conquer this one together!

Step 1: Unveiling the Greatest Common Factor (GCF) in $90x^2+60x-80$

Alright, guys, before we jump into the deeper factoring methods, the absolute first and most critical step when you're faced with any quadratic expression, especially one with bigger numbers like $90x^2+60x-80$, is to always look for the Greatest Common Factor (GCF). Seriously, don't skip this! Finding the GCF can dramatically simplify your problem, making the subsequent steps much, much easier to handle. Think of it like tidying up your workspace before a big project—it just makes everything flow better. The GCF is the largest number (and/or variable) that divides evenly into all the terms in your expression. In our case, we're looking at the numbers 90, 60, and -80. We need to find the biggest number that can go into all three of those without leaving a remainder. Let's break it down:

1. List the factors for each number (or use prime factorization):

   • Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
   • Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
   • Factors of 80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80

   Now, look at those lists. What's the largest number that appears in all three? If you scan through, you'll spot 10. Bingo! Ten is the GCF of 90, 60, and 80. Alternatively, you could use prime factorization: $90 = 2 imes 3^2 imes 5$, $60 = 2^2 imes 3 imes 5$, $80 = 2^4 imes 5$. The common prime factors with the lowest powers are $2^1$ and $5^1$, so $2 imes 5 = 10$. Easy peasy!

2. Factor out the GCF from each term: Once you've identified the GCF, you're going to divide each term in your original expression by that GCF. Remember, we're working with $90x^2+60x-80$.

   • $90x^2 ext{ divided by } 10 = 9x^2$
   • $60x ext{ divided by } 10 = 6x$
   • $-80 ext{ divided by } 10 = -8$

   So, after pulling out the GCF of 10, our expression transforms into: $10(9x^2+6x-8)$. See how much cleaner that looks? We've gone from big, intimidating numbers to something much more manageable inside the parentheses. This is a crucial step that many folks overlook, but it's your secret weapon for simplifying the entire factoring process. Always, always check for that GCF first! It saves a ton of headache down the line. We can't factor out any variables because the last term, -80, doesn't have an 'x'. So, 10 is our complete GCF for this expression. Now that we have a simpler quadratic inside, we're ready for the next big step: factoring that trinomial!

Step 2: Conquering the Quadratic: Factoring $9x^2+6x-8$

Alright, guys, now that we've expertly extracted the GCF, we're left with a much more approachable quadratic expression: $9x^2+6x-8$. This is a trinomial of the form $ax^2+bx+c$, where $a=9$, $b=6$, and $c=-8$. Since our 'a' value isn't 1, we can't just jump to finding two numbers that multiply to 'c' and add to 'b'. Instead, we're going to use a powerful technique called the AC Method (or sometimes called the Grouping Method). This method is super reliable for factoring these trickier quadratics.

Here's how it works for $9x^2+6x-8$:

1. Find the product of 'a' and 'c': This is where the "AC" in AC Method comes from. Multiply your 'a' term (9) by your 'c' term (-8). So, $9 imes (-8) = -72$. This is our magic number we're aiming for in multiplication.

2. Find two numbers that multiply to 'ac' and add to 'b': We need two numbers that, when multiplied together, give us -72, and when added together, give us our 'b' term, which is 6. This is the part that might require a little trial and error, but with practice, you'll get quicker at it. Let's list some pairs of factors for -72 and see which one sums to 6:

   • -1 and 72 (Sum: 71)
   • 1 and -72 (Sum: -71)
   • -2 and 36 (Sum: 34)
   • 2 and -36 (Sum: -34)
   • -3 and 24 (Sum: 21)
   • 3 and -24 (Sum: -21)
   • -4 and 18 (Sum: 14)
   • 4 and -18 (Sum: -14)
   • -6 and 12 (Sum: 6) Aha! We found our pair!
   • 6 and -12 (Sum: -6)

   Our two numbers are 12 and -6. They multiply to -72 and add up to 6.

3. Rewrite the middle term using these two numbers: Now, this is where the grouping part comes in. We're going to take our original middle term, $6x$, and split it into $12x - 6x$. This doesn't change the value of the expression, just its appearance, which is exactly what we need for factoring. So, $9x^2+6x-8$ becomes $9x^2 + 12x - 6x - 8$.

4. Group the terms and factor out the GCF from each pair: Now we have four terms, which means we can group them into two pairs. Let's group the first two terms and the last two terms:

   • $(9x^2 + 12x)$ and $(-6x - 8)$

   Next, find the GCF for each of these pairs:

   • For $(9x^2 + 12x)$, the GCF is $3x$. Factoring that out gives us $3x(3x + 4)$.
   • For $(-6x - 8)$, the GCF is $-2$. (Always pull out a negative if the first term in the group is negative to ensure your binomials match!) Factoring out $-2$ gives us $-2(3x + 4)$.

   Notice something awesome here? Both sets of parentheses contain the exact same binomial: $(3x+4)$! If they don't match, you've likely made a small error, so go back and check your numbers or your GCFs for the groups. This matching binomial is key to the next step.

5. Factor out the common binomial: Since both terms now share $(3x+4)$, we can factor that entire binomial out as a common factor. What's left over from each term is $(3x)$ from the first group and $(-2)$ from the second group. So, combining them gives us: $(3x+4)(3x-2)$.

And voilà! We've successfully factored the quadratic part of our original expression. This means $9x^2+6x-8$ is equal to $(3x+4)(3x-2)$. This is a crucial step, and mastering the AC method will open up so many doors in your algebraic journey. Keep practicing this technique, and you'll be a pro in no time!

Step 3: Bringing It All Together: The Complete Factorization

Alright, my factoring champions, we're in the home stretch! We've done the hard work of finding the GCF and then meticulously factoring the resulting quadratic expression. Now, it's time to put all the pieces back together to get our completely factored form of $90x^2+60x-80$. This step is super straightforward, but it's where many people accidentally leave off an important part of their answer, so pay close attention!

Think back to Step 1, where we identified the Greatest Common Factor (GCF) of 10 from our original expression. We started with $90x^2+60x-80$ and transformed it into $10(9x^2+6x-8)$. That '10' out front is still a part of the expression; we just temporarily set it aside to simplify the factoring of the trinomial inside the parentheses. It's like taking off the wrapper to get to the candy inside – you can't just throw the wrapper away, it's still part of the packaging if you want to represent the whole thing!

Then, in Step 2, we took that simpler quadratic, $9x^2+6x-8$, and through the awesome power of the AC method, we factored it into its two binomial factors: $(3x+4)(3x-2)$.

Now, all we have to do is reintroduce that GCF we found at the very beginning. We simply place it back in front of our newly factored binomials. So, the completely factored form of our original expression, $90x^2+60x-80$, becomes:

$10(3x+4)(3x-2)$

And there it is! You've officially factored the expression completely. This final answer means that if you were to multiply $10 imes (3x+4) imes (3x-2)$, you would end up right back with $90x^2+60x-80$. This is a fantastic way to check your work and ensure you haven't made any slip-ups along the way. Seriously, guys, take a moment to do a quick mental (or even written) check. Multiply $(3x+4)$ and $(3x-2)$ first using the FOIL method (First, Outer, Inner, Last): $(3x)(3x) + (3x)(-2) + (4)(3x) + (4)(-2) = 9x^2 - 6x + 12x - 8 = 9x^2 + 6x - 8$. Looks good! Then, distribute the 10: $10(9x^2+6x-8) = 90x^2+60x-80$. Perfect! The numbers match, the signs are correct, and everything lines up. This verification step is your ultimate safety net and solidifies your understanding. Remember, the goal of factoring completely is to break down the expression into its simplest, irreducible multiplicative components. By including the GCF, you've ensured that every piece of the original puzzle is accounted for. Give yourselves a pat on the back, you've nailed it!

Practical Tips and Common Mistakes When Factoring Quadratics

Alright, folks, you've just conquered a pretty chunky quadratic expression in $90x^2+60x-80$, which is awesome! But like with any skill, there are some common pitfalls and best practices that can make your life a whole lot easier when factoring other quadratics. Learning from these tips and understanding potential mistakes will not only speed up your factoring process but also boost your accuracy significantly. So, let's chat about a few things to keep in mind:

1. Always, Always, ALWAYS Check for the GCF First! I can't stress this enough. As we saw with $90x^2+60x-80$, finding the Greatest Common Factor (GCF) of 10 right at the beginning simplified the problem immensely. Trying to factor $90x^2+60x-80$ directly using the AC method (multiplying $90 imes -80 = -7200$) would be an absolute nightmare, involving massive numbers and making it much harder to find the correct pair. Always start by scanning all terms for common factors. If you forget, you might still get a correct factorization, but it won't be completely factored, and your numbers will be much larger and more unwieldy.

2. Be Super Careful with Signs: This is where many students trip up. A misplaced negative sign can completely derail your factoring. When using the AC method, pay close attention to whether 'ac' is positive or negative. If 'ac' is positive, your two numbers will either both be positive (if 'b' is positive) or both be negative (if 'b' is negative). If 'ac' is negative (like our -72), one number must be positive and the other negative. Similarly, when factoring by grouping, ensure you factor out the correct sign, especially when the leading term of a group is negative. For instance, in $(-6x-8)$, factoring out $-2$ correctly yielded $(3x+4)$, which matched our other binomial. If you had factored out just $2$, you would have gotten $(-3x-4)$, which doesn't match and causes issues.

3. Don't Forget the GCF in Your Final Answer: After doing all that hard work, it's really easy to just write down the factored binomials, like $(3x+4)(3x-2)$, and forget about that GCF you pulled out at the start. Remember, the GCF is an essential multiplier for the entire expression. Our final, completely factored answer for $90x^2+60x-80$ must include the '10': $10(3x+4)(3x-2)$. Leaving it out means your answer isn't equivalent to the original expression.

4. Practice Makes Perfect (and Faster!): Factoring is a skill, and like any skill, it improves with practice. The more you work through different types of quadratic expressions, the quicker you'll become at recognizing patterns, finding GCFs, and identifying the correct number pairs for the AC method. Don't get discouraged if it feels slow at first; everyone starts there. Grab some extra practice problems and just keep at it!

5. Always Check Your Work by Multiplying Back Out: This is your secret weapon, guys! As we demonstrated earlier, if you multiply your factored form back together, you should arrive at the original expression. This takes a few extra seconds but provides instant verification that your factoring is correct. For $10(3x+4)(3x-2)$, if you multiply the binomials first to get $9x^2+6x-8$, and then distribute the 10, you get $90x^2+60x-80$. If it doesn't match, you know exactly where to start looking for your mistake.

By keeping these practical tips and common pitfalls in mind, you'll not only master expressions like $90x^2+60x-80$ but also build a robust foundation for all your future algebraic endeavors. You've got this!

Beyond the Basics: Where Factoring Takes You

So, guys, we've successfully navigated the exciting world of factoring completely by breaking down $90x^2+60x-80$ into its irreducible components: $10(3x+4)(3x-2)$. That's a huge achievement! But I want you to remember that learning to factor isn't just about solving one specific problem. It's about equipping yourself with a fundamental tool that unlocks doors to so many other fascinating areas in mathematics and beyond. Think of factoring as learning the alphabet before you can write a novel; it's a foundational skill that enables far more complex and interesting work. When you truly grasp how to factor completely, you gain a deeper insight into the structure of polynomial expressions, understanding how they're built and how their individual pieces contribute to the whole. This conceptual understanding is incredibly valuable.

For instance, the immediate next step for many students after mastering factoring is using it to solve quadratic equations. If you were given the equation $90x^2+60x-80=0$, you could now confidently rewrite it as $10(3x+4)(3x-2)=0$. From there, the Zero Product Property becomes your best friend! It tells us that if a product of factors is zero, then at least one of those factors must be zero. So, you'd set $3x+4=0$ and $3x-2=0$ (we can ignore the 10 since $10 eq 0$). Solving these simple linear equations gives you $x = -4/3$ and $x = 2/3$. These are the roots of the quadratic equation, and they tell you exactly where the graph of the parabola $y = 90x^2+60x-80$ crosses the x-axis. Pretty neat, right?

Beyond solving equations, factoring is indispensable in calculus for finding critical points, in physics for modeling trajectories and forces, in engineering for designing structures, and even in computer science for optimizing algorithms. Simplifying rational expressions (fractions with polynomials) by factoring numerators and denominators is a common task. Moreover, the conceptual understanding gained from factoring builds a strong foundation for understanding higher-degree polynomials and their roots. It helps you see connections and patterns that might otherwise remain hidden. So, as you continue your mathematical journey, remember that the skills you honed today with $90x^2+60x-80$ will continue to serve you well, opening up new avenues for problem-solving and deeper mathematical exploration. Don't stop here; keep practicing, keep questioning, and keep exploring, because the world of math is full of incredible discoveries waiting for you!