Factoring $15x^2 + 19xy + 6y^2$: A Step-by-Step Guide

Hey guys! Today, we're diving into factoring a quadratic expression with two variables. Specifically, we're going to tackle the expression $15x^2 + 19xy + 6y^2$. Factoring can seem daunting at first, but with a systematic approach, it becomes much more manageable. So, grab your pencils and let's get started!

Understanding the Problem

Before we jump into the factoring process, let's understand what we're dealing with. We have a quadratic expression in two variables, $x$ and $y$. The general form of such an expression is $ax^2 + bxy + cy^2$, where $a$, $b$, and $c$ are constants. In our case, $a = 15$, $b = 19$, and $c = 6$. Our goal is to rewrite this expression as a product of two binomials, like this: $(px + qy)(rx + sy)$, where $p$, $q$, $r$, and $s$ are also constants. The challenge is finding the right values for these constants.

Factoring quadratic expressions is a fundamental skill in algebra, and mastering it opens doors to solving more complex equations and simplifying expressions. It's like having a secret weapon in your mathematical arsenal! The expression $15x^2 + 19xy + 6y^2$ might look intimidating, but don't worry; we'll break it down step by step. Essentially, what we're trying to do is reverse the process of expanding two binomials. Remember how the FOIL method (First, Outer, Inner, Last) helps us multiply binomials? Well, factoring is like going backward from the result of the FOIL method to find the original binomials.

Understanding the coefficients is crucial. The coefficient 'a' (15 in this case) is associated with the $x^2$ term, 'b' (19) with the $xy$ term, and 'c' (6) with the $y^2$ term. These numbers hold the key to unlocking the factored form. The $xy$ term is what makes this slightly trickier than a simple quadratic in one variable, but the same principles apply. We need to find two binomials that, when multiplied, give us these exact coefficients. This involves a bit of trial and error, but don't be discouraged! With practice, you'll develop an intuition for what numbers to look for.

Why is factoring important anyway? Factoring isn't just an abstract mathematical exercise. It's a powerful tool used in various fields, from engineering and physics to computer science and economics. For example, in physics, factoring can help simplify equations related to motion and energy. In computer science, it's used in cryptography and data compression. By mastering factoring, you're not just learning a math skill; you're gaining a valuable problem-solving tool that can be applied in many real-world scenarios.

Step-by-Step Factoring Process

Here's how we can factor the expression $15x^2 + 19xy + 6y^2$:

1. Identify $a$, $b$, and $c$:

   • As we mentioned earlier, $a = 15$, $b = 19$, and $c = 6$.

2. Calculate $ac$:

   • Multiply $a$ and $c$: $15 \times 6 = 90$.

3. Find two numbers that multiply to $ac$ (90) and add up to $b$ (19):

   • We need to find two numbers that when multiplied equal 90, and when added equal 19. Let's list the factor pairs of 90:
     • 1 and 90
     • 2 and 45
     • 3 and 30
     • 5 and 18
     • 6 and 15
     • 9 and 10
   • Looking at these pairs, we see that 9 and 10 satisfy our condition because $9 ") 10 = 90$ and $9 + 10 = 19$.

4. Rewrite the middle term ($19xy$) using the two numbers we found (9 and 10):

   • $15x^2 + 19xy + 6y^2 = 15x^2 + 9xy + 10xy + 6y^2$

5. Factor by grouping:

   • Group the first two terms and the last two terms:
     • $(15x^2 + 9xy) + (10xy + 6y^2)$
   • Factor out the greatest common factor (GCF) from each group:
     • From the first group, the GCF is $3x$: $3x(5x + 3y)$
     • From the second group, the GCF is $2y$: $2y(5x + 3y)$
   • Now we have: $3x(5x + 3y) + 2y(5x + 3y)$
   • Notice that $(5x + 3y)$ is a common factor in both terms. Factor it out:
     • $(5x + 3y)(3x + 2y)$

6. Check your answer by expanding the factored form:

   • $(5x + 3y)(3x + 2y) = 5x(3x) + 5x(2y) + 3y(3x) + 3y(2y) = 15x^2 + 10xy + 9xy + 6y^2 = 15x^2 + 19xy + 6y^2$
   • Our factored form is correct!

Step 1: Identifying a, b, and c is like setting the stage for a play. You need to know who the main characters are before the story can unfold. In this case, a, b, and c are the coefficients that define our quadratic expression. They're the building blocks upon which we'll construct our factored form. Make sure you identify them correctly, as a mistake here can throw off the entire process.

Step 2: Calculating ac might seem like a random step, but it's actually a clever trick that helps us find the right numbers for factoring. By multiplying 'a' and 'c', we're creating a new number that encapsulates the relationship between the first and last terms of the quadratic expression. This number becomes our target product when we're looking for two numbers that satisfy our conditions.

Step 3: Finding two numbers that multiply to ac and add up to b is the heart of the factoring process. It's like solving a puzzle where you need to find two pieces that fit together perfectly. This step requires a bit of trial and error, but with practice, you'll develop an intuition for what numbers to look for. Listing out the factor pairs of 'ac' can be a helpful strategy, as it allows you to systematically explore the possibilities.

Step 4: Rewriting the middle term is where we start to transform our original expression into a form that's easier to factor. By splitting the middle term ('bxy') into two terms using the numbers we found in the previous step, we're essentially creating two pairs of terms that share common factors. This sets us up for the next step, which is factoring by grouping.

Step 5: Factoring by grouping is a powerful technique that allows us to extract common factors from pairs of terms. By grouping the first two terms and the last two terms, we can identify common factors within each group. Factoring out these common factors reveals a shared binomial factor, which we can then factor out again to obtain the final factored form. This step is like peeling back the layers of an onion to reveal the core structure.

Step 6: Checking your answer is a crucial step that ensures you haven't made any mistakes along the way. By expanding the factored form, you can verify that it's equivalent to the original expression. This step is like proofreading your work to catch any errors before submitting it.

Example

Let's do another quick example to solidify our understanding. Factor the expression $6x^2 + 13x + 5$.

1. $a = 6$, $b = 13$, $c = 5$
2. $ac = 6 \times 5 = 30$
3. Find two numbers that multiply to 30 and add up to 13: 3 and 10
4. Rewrite the middle term: $6x^2 + 3x + 10x + 5$
5. Factor by grouping: $(6x^2 + 3x) + (10x + 5) = 3x(2x + 1) + 5(2x + 1) = (2x + 1)(3x + 5)$
6. Check: $(2x + 1)(3x + 5) = 6x^2 + 10x + 3x + 5 = 6x^2 + 13x + 5$. Correct!

Remember, practice makes perfect! The more you practice factoring quadratic expressions, the easier it will become. Don't be afraid to make mistakes; they're part of the learning process. The key is to learn from your mistakes and keep practicing until you're comfortable with the process.

Conclusion

So, there you have it! We've successfully factored the quadratic expression $15x^2 + 19xy + 6y^2$ into $(5x + 3y)(3x + 2y)$. Factoring can be a bit tricky at first, but with practice and a systematic approach, you can master it. Keep practicing, and you'll be factoring like a pro in no time! Remember the steps: identify, calculate, find the numbers, rewrite, group, and check. Happy factoring, everyone! You've got this!

By following these steps, you'll be able to tackle a wide variety of quadratic expressions. Keep practicing and don't be afraid to ask for help when you need it. Factoring is a valuable skill that will serve you well in your mathematical journey. So go out there and conquer those quadratic expressions! Good luck, and have fun!