Factoring The Quadratic Expression X² - 8xy + 15y² A Comprehensive Guide

Jul 13, 2025 by ADMIN 73 views

Factoring quadratic expressions is a fundamental skill in algebra. It's the reverse process of expanding brackets, and it's essential for solving equations, simplifying expressions, and understanding the behavior of functions. One common type of quadratic expression involves two variables, like the one we're going to tackle: x² - 8xy + 15y². This expression looks a bit more complex than the standard ax² + bx + c form because of the xy term, but the underlying principles of factoring remain the same. In this detailed explanation, we will break down the steps required to factor this expression, discuss the logic behind each step, and provide insights into why this method works. Understanding these concepts will empower you to confidently tackle similar factoring problems in the future. The correct answer of the provided question is D. $(x - 5y)(x - 3y)$ , let's delve into a comprehensive explanation of how we arrive at this solution.

Understanding the Problem

Before diving into the solution, let's first understand what we're trying to achieve. Factoring a quadratic expression means rewriting it as a product of two binomials. In simpler terms, we want to find two expressions in the form (ax + by)(cx + dy) that, when multiplied together, give us the original expression, x² - 8xy + 15y². Recognizing the structure of the quadratic expression is the first step. We have a term with x², a term with xy, and a term with y². This suggests that the binomial factors will likely involve terms with x and y. The coefficients in the expression (1, -8, and 15) will guide us in finding the correct combination of numbers for our binomials. To master factoring, it's crucial to recognize patterns and relationships between the coefficients and the resulting factors. This skill is not only useful for algebra but also for more advanced mathematical concepts. The ability to quickly and accurately factor expressions can significantly simplify problem-solving in calculus, trigonometry, and other areas of mathematics.

The Factoring Process: A Step-by-Step Guide

To factor the quadratic expression x² - 8xy + 15y², we'll follow these steps:

1. Identify the Coefficients

First, let's identify the coefficients of each term in the expression. We have:

Coefficient of x²: 1
Coefficient of xy: -8
Coefficient of y²: 15

These coefficients are the key to unlocking the factors. The coefficient of x² tells us about the leading terms in our binomials, while the coefficients of xy and y² provide clues about the constant terms. In essence, factoring is like solving a puzzle where we need to find the right pieces (the numbers) that fit together to form the original picture (the expression). The coefficients give us the dimensions and constraints of the puzzle, making it easier to find the solution. Understanding the role of each coefficient is essential for efficient factoring. Without this understanding, the process can feel like guesswork. However, with a clear grasp of the relationships between coefficients and factors, you can approach factoring with confidence and precision.

2. Find Two Numbers

We need to find two numbers that:

Multiply to the coefficient of y² (15)
Add up to the coefficient of xy (-8)

This is the core of the factoring process. We're looking for two numbers that satisfy two conditions simultaneously. This is where your knowledge of number properties and your ability to recognize patterns come into play. We need to consider both positive and negative factors since the product is positive (15) and the sum is negative (-8). This suggests that both numbers must be negative. Thinking about the factor pairs of 15, we have (1, 15) and (3, 5). Since we need a negative sum, we consider (-1, -15) and (-3, -5). The pair (-3, -5) satisfies both conditions: -3 multiplied by -5 equals 15, and -3 plus -5 equals -8. This step often involves some trial and error, but with practice, you'll develop a sense for which numbers are likely to work. The key is to systematically consider the factors of the constant term and check if their sum matches the coefficient of the middle term.

3. Construct the Factors

Once we've found the two numbers (-3 and -5), we can construct the factors. These numbers will be the constant terms in our binomials. Since the coefficient of x² is 1, the leading terms in our binomials will simply be x. We can now write the factors as:

(x - 3y)(x - 5y)

Notice how we've incorporated the y term. This is because the original expression has terms with y, specifically xy and y². The y in the binomials ensures that when we multiply them out, we get these terms. The structure of the factors directly reflects the structure of the original expression. The x terms account for the x² term, the constant terms (-3y and -5y) account for the 15y² term, and the combination of x and y terms creates the -8xy term. Building the factors in this way is a logical and systematic approach that minimizes errors and ensures that you arrive at the correct solution. The ability to construct factors accurately is a cornerstone of algebraic manipulation.

4. Verify the Solution

To make sure we've factored correctly, we can expand the factors and see if we get back the original expression:

(x - 3y)(x - 5y) = x² - 5xy - 3xy + 15y² = x² - 8xy + 15y²

Since the expanded form matches the original expression, our factoring is correct. Verification is a crucial step in any mathematical problem. It's a way to double-check your work and ensure that you haven't made any mistakes. In this case, expanding the factors is a straightforward process that allows us to confirm that our factoring is accurate. This step provides peace of mind and reinforces the connection between factoring and expanding. The more you practice this verification step, the more confident you'll become in your factoring abilities. Moreover, this process helps to solidify your understanding of the distributive property and how it applies to binomial multiplication.

Why This Method Works: A Deeper Look

The method we used works because of the distributive property of multiplication over addition. When we expand the product of two binomials, we're essentially applying the distributive property multiple times. Let's break down the expansion of (x - 3y)(x - 5y) to see this in action:

Distribute the first term of the first binomial (x) over the second binomial: x(x - 5y) = x² - 5xy
Distribute the second term of the first binomial (-3y) over the second binomial: -3y(x - 5y) = -3xy + 15y²
Combine the results: x² - 5xy - 3xy + 15y² = x² - 8xy + 15y²

Notice how the -5xy and -3xy terms combine to give us the -8xy term in the original expression. This is why we needed to find two numbers that add up to -8. Similarly, the product of the constant terms (-3y and -5y) gives us the 15y² term, which is why we needed to find two numbers that multiply to 15. The factoring process is essentially the reverse of this expansion. We're trying to undo the distributive property and find the two binomials that, when multiplied, give us the original expression. Understanding this connection between factoring and expanding is crucial for mastering algebraic manipulation. It provides a deeper insight into the underlying principles and allows you to approach factoring problems with a more strategic mindset. Instead of blindly following steps, you can understand why each step is necessary and how it contributes to the overall solution.

Common Mistakes to Avoid

Factoring can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to watch out for:

Incorrect Signs: Pay close attention to the signs of the coefficients. A mistake in the sign can lead to completely incorrect factors. For instance, if you incorrectly identify the signs, you might end up with factors like (x + 3y)(x + 5y), which expands to x² + 8xy + 15y², a different expression altogether. Double-checking the signs is a simple but crucial step in the factoring process.
Forgetting the 'y' Term: When factoring expressions with two variables, remember to include the 'y' term in the binomials. This is a common oversight, especially for beginners. If you forget the 'y', your factors won't correctly account for the xy and y² terms in the original expression. Remember that the structure of the factors must mirror the structure of the expression being factored.
Incorrectly Identifying Factors: Make sure you find the correct pair of numbers that multiply to the coefficient of y² and add up to the coefficient of xy. This is the heart of the factoring process, and any mistake here will lead to incorrect factors. It's helpful to systematically list out the factor pairs of the constant term and check their sums to avoid errors.
Not Verifying the Solution: Always verify your solution by expanding the factors. This is the best way to catch any mistakes you might have made. Expanding the factors is a straightforward process that allows you to confirm that you've factored correctly and provides peace of mind.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in factoring. Remember that practice is key to mastering any mathematical skill, and factoring is no exception. The more you practice, the more comfortable and proficient you'll become.

Practice Problems

To solidify your understanding of factoring, try these practice problems:

Factor x² - 10xy + 21y²
Factor x² + 2xy - 15y²
Factor x² - 4xy - 12y²

Working through these problems will help you apply the steps we've discussed and identify any areas where you might need more practice. Factoring is a skill that builds upon itself, so the more you practice, the better you'll become. Don't be afraid to make mistakes – they're a natural part of the learning process. The key is to learn from your mistakes and continue to practice until you've mastered the skill. Remember to always verify your solutions by expanding the factors. This will not only help you catch any errors but also reinforce your understanding of the factoring process.

Conclusion

Factoring the quadratic expression x² - 8xy + 15y² involves finding two binomials that, when multiplied, give us the original expression. By identifying the coefficients, finding the correct numbers, constructing the factors, and verifying the solution, we can confidently factor this type of expression. The factors of x² - 8xy + 15y² are indeed (x - 5y)(x - 3y). Factoring quadratic expressions is a fundamental skill in algebra, with applications in various areas of mathematics and beyond. Mastering this skill requires a solid understanding of the underlying principles, a systematic approach, and consistent practice. By following the steps outlined in this article and working through practice problems, you can develop your factoring abilities and confidently tackle a wide range of algebraic problems. Remember that mathematics is a journey of learning and discovery, and every problem you solve brings you one step closer to mastery. Embrace the challenge, persevere through difficulties, and celebrate your successes along the way. With dedication and effort, you can unlock the power of algebra and its countless applications.