Simplifying The Expression (-5-√3)^2 A Step-by-Step Guide

Introduction

In this article, we will delve into simplifying the expression $(-5-\sqrt{3})^2$. This task involves expanding a binomial expression, which is a fundamental concept in algebra. Mastering these simplification techniques is crucial for solving more complex mathematical problems. Our goal is to break down the steps in a clear and detailed manner, ensuring that you understand not just the how but also the why behind each step. We'll cover the necessary algebraic rules and demonstrate their application in a way that’s easy to follow. Whether you're a student looking to improve your algebra skills or simply someone who enjoys mathematical challenges, this guide will provide you with a comprehensive understanding of how to simplify such expressions. Let's embark on this mathematical journey and unravel the solution together.

Understanding the Basics: Expanding Binomials

Before we jump into the specific problem, let's establish the foundational principle we'll be using: the expansion of a binomial. A binomial is an algebraic expression with two terms, and when we square a binomial, we're essentially multiplying it by itself. The general formula for expanding $(a + b)^2$ is given by $(a + b)^2 = a^2 + 2ab + b^2$. This formula arises from the distributive property of multiplication over addition, often remembered by the acronym FOIL (First, Outer, Inner, Last) when multiplying two binomials. To simplify $(-5-\sqrt{3})^2$, we can apply this formula with $a = -5$ and $b = -\sqrt{3}$. Understanding this basic formula is key to accurately simplifying more complex expressions. The expansion process not only involves squaring each term individually but also includes the crucial step of multiplying the two terms and doubling the result. This ensures that we account for all possible combinations when the binomial is multiplied by itself. Let’s now apply this understanding to our specific expression and see how it unfolds step by step.

Step-by-Step Simplification of $(-5-\sqrt{3})^2$

Now, let’s apply the binomial expansion formula to simplify $(-5-\sqrt{3})^2$. We'll take it step by step to ensure clarity. Recall that our formula is $(a + b)^2 = a^2 + 2ab + b^2$, where $a = -5$ and $b = -\sqrt{3}$.

1. Square the first term ($a^2$): $(-5)^2 = 25$. This is straightforward: multiplying -5 by itself gives us 25.
2. Multiply the two terms and double the result ($2ab$): $2(-5)(-\sqrt{3}) = 10\sqrt{3}$. Here, we multiply -5 and -\sqrt{3} to get $5\sqrt{3}$, and then double it to obtain $10\sqrt{3}$.
3. Square the second term ($b^2$): $(-\sqrt{3})^2 = 3$. Squaring a negative square root results in the positive integer under the radical.
4. Combine the results: Now, we add the results from the previous steps: $25 + 10\sqrt{3} + 3$.
5. Simplify by combining like terms: $25 + 3 = 28$, so our expression becomes $28 + 10\sqrt{3}$.

Thus, $(-5-\sqrt{3})^2$ simplifies to $28 + 10\sqrt{3}$. This methodical approach ensures that we cover all components of the expansion and arrive at the correct simplified form. Understanding each step not only helps in solving this particular problem but also builds a strong foundation for tackling other algebraic challenges. Let’s now analyze the result and see how it aligns with the given options.

Analyzing the Result and Comparing with Options

Having simplified $(-5-\sqrt{3})^2$ to $28 + 10\sqrt{3}$, we now need to verify this result against the provided options. The options given were:

• $28-10 \sqrt{3}$
• $-13+5 \sqrt{3}$
• $25-10 \sqrt{3}$
• $28+10 \sqrt{3}$

Our simplified expression, $28 + 10\sqrt{3}$, directly matches the fourth option. This confirms that our step-by-step simplification process was accurate. It's essential to compare your final result with the given options to ensure that no errors were made during the calculation. Matching the result with one of the options provides confidence in the solution. Furthermore, if our result didn't match any of the options, it would indicate a need to revisit the steps and identify any potential mistakes. In this case, the clear match reinforces our understanding and execution of the binomial expansion. Let's now discuss some common pitfalls to avoid when simplifying such expressions.

Common Mistakes to Avoid

When simplifying expressions like $(-5-\sqrt{3})^2$, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them. One frequent error is incorrectly applying the binomial expansion formula. For instance, some might forget to include the middle term ($2ab$) or miscalculate it. It's crucial to remember that $(a + b)^2$ is not equal to $a^2 + b^2$; the $2ab$ term is essential for the correct expansion. Another mistake is mishandling the negative signs. In our expression, both terms inside the parentheses are negative, and incorrectly squaring or multiplying them can lead to a sign error in the final result. For example, $(-5)^2$ is 25, not -25. Similarly, when multiplying $2(-5)(-\sqrt{3})$, the negative signs cancel out, resulting in a positive term. Errors in arithmetic, such as incorrect multiplication or addition, can also occur, especially when dealing with square roots. Always double-check your calculations, particularly when combining like terms. Lastly, it’s vital to simplify the square root term correctly. Remember that $(\sqrt{3})^2$ is simply 3, as squaring a square root cancels out the radical. By being mindful of these common mistakes and practicing careful calculation, you can increase your accuracy in simplifying algebraic expressions.

Alternative Approaches to Simplification

While we've primarily focused on using the binomial expansion formula, there are alternative approaches to simplifying $(-5-\sqrt{3})^2$ that can provide additional insight and reinforce understanding. One such approach is direct multiplication. Instead of applying the formula, we can write $(-5-\sqrt{3})^2$ as $(-5-\sqrt{3})(-5-\sqrt{3})$ and then use the distributive property (or the FOIL method) to multiply the two binomials. This involves multiplying each term in the first binomial by each term in the second binomial and then combining like terms. This method is particularly useful for those who prefer a more step-by-step approach or want to visually ensure that every term is accounted for. Another approach involves recognizing the structure of the expression and breaking it down into smaller, more manageable parts. For instance, one could first simplify the expression inside the parentheses, if possible, and then proceed with squaring the result. While this might not always be applicable, it can be beneficial in certain scenarios. Exploring these alternative methods not only provides a deeper understanding of the problem but also enhances problem-solving skills by offering different perspectives and techniques. Whether you choose to use the binomial expansion formula or direct multiplication, the key is to select the method that you find most comfortable and accurate.

Conclusion

In conclusion, we have successfully simplified the expression $(-5-\sqrt{3})^2$ to $28 + 10\sqrt{3}$. We began by understanding the basic principles of binomial expansion, applied the formula $(a + b)^2 = a^2 + 2ab + b^2$, and carefully executed each step. We squared the first term, multiplied the two terms and doubled the result, squared the second term, and combined the results, ensuring accuracy throughout the process. Our step-by-step approach not only led us to the correct answer but also provided a clear understanding of the methodology involved. We then verified our result against the given options, confirming the accuracy of our solution. Additionally, we discussed common mistakes to avoid when simplifying such expressions, such as misapplying the binomial expansion formula, mishandling negative signs, and errors in arithmetic. Understanding these potential pitfalls can significantly improve accuracy in algebraic manipulations. Finally, we explored alternative approaches to simplification, including direct multiplication, highlighting the importance of having multiple problem-solving strategies. By mastering these techniques, you can confidently tackle similar algebraic challenges and enhance your mathematical skills. Simplifying expressions is a fundamental skill in algebra, and with practice and a clear understanding of the principles involved, it becomes a manageable and rewarding task.