Expanding Algebraic Expressions Solving -5x(2x + 6y)

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Expanding algebraic expressions is a fundamental skill in mathematics, essential for simplifying equations, solving problems, and understanding more advanced concepts. In this article, we will delve into the process of expanding expressions, focusing on the specific example of -5x(2x + 6y). We will break down the steps involved, explain the underlying principles, and provide a clear, comprehensive guide to help you master this important skill. Whether you're a student just starting out with algebra or someone looking to refresh your knowledge, this article will provide you with the tools and understanding you need to confidently tackle expanding algebraic expressions.

Understanding the Distributive Property

The distributive property is the cornerstone of expanding algebraic expressions. This property states that multiplying a single term by an expression inside parentheses is the same as multiplying the term by each individual term within the parentheses and then adding or subtracting the results. In mathematical terms, this can be expressed as: a(b + c) = ab + ac. This seemingly simple rule is incredibly powerful and allows us to transform complex expressions into simpler, more manageable forms. To truly grasp the distributive property, it's helpful to visualize it. Imagine you have a group of objects, say 5 boxes, and each box contains 2 apples and 3 oranges. To find the total number of apples, you could add the apples in each box (2 + 2 + 2 + 2 + 2 = 10) or you could multiply the number of boxes by the number of apples in each box (5 * 2 = 10). The same principle applies to oranges (5 * 3 = 15). The distributive property essentially allows us to perform these calculations in a similar way with algebraic expressions.

When working with algebraic expressions, the distributive property becomes even more crucial. It allows us to remove parentheses and combine like terms, which is essential for solving equations and simplifying expressions. For example, consider the expression 3(x + 4). Using the distributive property, we multiply 3 by both x and 4, resulting in 3x + 12. This transformation makes the expression easier to work with and allows us to further simplify it if necessary. The distributive property is not limited to simple binomials (expressions with two terms) like (x + 4). It can be applied to expressions with any number of terms, such as 2(a + b + c) or even more complex expressions. The key is to remember to multiply the term outside the parentheses by every term inside the parentheses.

Applying the Distributive Property to -5x(2x + 6y)

Let's apply the distributive property to our specific expression: -5x(2x + 6y). This expression involves multiplying the term -5x by the binomial (2x + 6y). To do this, we'll follow the distributive property and multiply -5x by each term inside the parentheses individually. First, we multiply -5x by 2x. Remember that when multiplying variables, we multiply the coefficients (the numbers in front of the variables) and add the exponents. In this case, -5 * 2 = -10 and x * x = x². So, the first term becomes -10x². Next, we multiply -5x by 6y. Again, we multiply the coefficients: -5 * 6 = -30. When multiplying different variables, we simply write them next to each other. So, x * y = xy. Therefore, the second term becomes -30xy. Now, we combine the two terms we obtained: -10x² and -30xy. This gives us the expanded expression: -10x² - 30xy. This is the simplified form of the original expression, and it represents the same value as the original expression for all values of x and y.

To further illustrate this process, let's break it down step-by-step:

  1. Identify the term outside the parentheses: In this case, it's -5x.
  2. Identify the terms inside the parentheses: Here, we have 2x and 6y.
  3. Multiply the term outside the parentheses by the first term inside: -5x * 2x = -10x²
  4. Multiply the term outside the parentheses by the second term inside: -5x * 6y = -30xy
  5. Combine the resulting terms: -10x² - 30xy

By following these steps, you can confidently expand any expression of this form. Remember to pay close attention to the signs (positive or negative) of the terms, as this is a common area for errors. Practice makes perfect, so try working through similar examples to solidify your understanding of the distributive property.

Step-by-Step Solution: -5x(2x + 6y)

Let's meticulously break down the expansion of -5x(2x + 6y) step-by-step to ensure clarity and understanding. This process involves careful application of the distributive property and attention to the rules of multiplying variables and their coefficients. By following each step, you'll gain a deeper understanding of how to expand algebraic expressions effectively.

  1. Distribute -5x to 2x:

    The first step is to multiply -5x by the first term inside the parentheses, which is 2x. When multiplying terms with variables, we multiply the coefficients and add the exponents of the variables. Remember that x is equivalent to x¹, so when we multiply x by x, we are essentially multiplying x¹ by x¹, which results in x¹⁺¹ = x².

    So, -5x * 2x = (-5 * 2) * (x * x) = -10x²

    Here, we multiplied the coefficients -5 and 2 to get -10, and we multiplied the variables x and x to get x². This first part of the expansion gives us the term -10x².

  2. Distribute -5x to 6y:

    Next, we multiply -5x by the second term inside the parentheses, which is 6y. This step is similar to the previous one, but we are now multiplying two different variables, x and y. When multiplying different variables, we simply write them next to each other.

    So, -5x * 6y = (-5 * 6) * (x * y) = -30xy

    In this case, we multiplied the coefficients -5 and 6 to get -30, and we multiplied the variables x and y to get xy. This gives us the second term of the expanded expression, -30xy.

  3. Combine the terms:

    The final step is to combine the terms we obtained in the previous two steps. We have -10x² and -30xy. Since these terms have different variable parts (x² and xy), they are not like terms and cannot be combined further. Therefore, the expanded expression is simply the sum of these two terms.

    -10x² - 30xy

    This is the final expanded form of the expression -5x(2x + 6y). We have successfully applied the distributive property and simplified the expression.

By following these steps, you can confidently expand similar algebraic expressions. Remember to pay close attention to the signs of the terms and the rules of multiplying variables. With practice, this process will become second nature.

Common Mistakes to Avoid

When expanding algebraic expressions, several common mistakes can lead to incorrect results. Being aware of these pitfalls can help you avoid them and ensure accurate calculations. Here, we'll discuss some of the most frequent errors and provide tips on how to prevent them. One of the most common mistakes is forgetting to distribute the term outside the parentheses to every term inside. For instance, in the expression -5x(2x + 6y), some might correctly multiply -5x by 2x but forget to multiply it by 6y. This omission leads to an incomplete and incorrect expansion. To avoid this, make a conscious effort to distribute the term to each term within the parentheses, perhaps even drawing arrows to visually connect the terms you're multiplying.

Another frequent error is related to the signs (positive or negative) of the terms. When multiplying a negative term by a positive term, the result is negative, and when multiplying a negative term by a negative term, the result is positive. For example, in our expression, -5x multiplied by 6y results in -30xy, not 30xy. A simple way to remember this is the rule: "same signs positive, different signs negative." Pay close attention to the signs and double-check your work to catch any errors. Mistakes can also occur when multiplying variables. Remember that when multiplying variables with exponents, you add the exponents. For example, x * x is x², not 2x. Similarly, when multiplying different variables, like x and y, you simply write them next to each other as xy. Avoid the common mistake of adding the variables or incorrectly combining them. Keeping the rules of exponents and variable multiplication in mind will help you avoid these errors.

Finally, another source of errors is combining unlike terms. Only terms with the exact same variable part can be combined. For example, -10x² and -30xy cannot be combined because one term has x² and the other has xy. Trying to combine them would be like trying to add apples and oranges – they are different and cannot be added together. Make sure to only combine terms that have the same variable and exponent. By being mindful of these common mistakes and practicing regularly, you can significantly improve your accuracy and confidence in expanding algebraic expressions.

Conclusion: Mastering Expansion for Algebraic Success

In conclusion, mastering the expansion of algebraic expressions is a crucial step towards success in algebra and beyond. The distributive property serves as the cornerstone of this skill, allowing us to simplify complex expressions and solve equations effectively. By diligently applying the distributive property, paying close attention to signs and variable multiplication, and avoiding common mistakes, you can confidently expand a wide range of algebraic expressions. The specific example we explored, -5x(2x + 6y), highlights the step-by-step process involved in expanding expressions. We saw how distributing -5x to both 2x and 6y led to the expanded form, -10x² - 30xy. This process reinforces the importance of multiplying each term inside the parentheses by the term outside, ensuring a complete and accurate expansion.

Furthermore, understanding the common mistakes to avoid, such as neglecting to distribute to all terms, mismanaging signs, incorrectly multiplying variables, and combining unlike terms, is essential for preventing errors. By being mindful of these pitfalls, you can significantly improve your accuracy and achieve correct results. The ability to expand algebraic expressions is not just a standalone skill; it's a foundational building block for more advanced algebraic concepts. Simplifying equations, solving inequalities, factoring polynomials, and even calculus all rely on a solid understanding of expansion. Therefore, investing the time and effort to master this skill will pay dividends throughout your mathematical journey.

To further enhance your understanding and proficiency, practice is key. Work through numerous examples, starting with simple expressions and gradually progressing to more complex ones. Seek out opportunities to apply expansion in different contexts, such as solving equations or simplifying formulas. The more you practice, the more natural and intuitive the process will become. Ultimately, mastering expansion is about developing a deep understanding of the underlying principles and applying them consistently and accurately. With dedication and practice, you can confidently expand algebraic expressions and unlock new levels of mathematical understanding and success.

Possible Answers

Based on our comprehensive step-by-step solution, the correct expansion of the expression -5x(2x + 6y) is:

-10x² - 30xy

Therefore, the correct answer from the options provided is:

(1) -10x² - 30xy

The other options are incorrect due to errors in applying the distributive property or mismanaging the signs during multiplication.