Multiply Complex Numbers: (6-5i)^2 Solution

Hey guys! Today, we're diving into the fascinating world of complex numbers and tackling a common problem: multiplying a complex number by itself. Specifically, we'll be working through the expression $(6 - 5i)^2$. Complex numbers might seem a bit intimidating at first, but trust me, once you grasp the basics, they're actually quite fun to work with. So, let's break it down step-by-step and make sure you understand each part of the process. We'll not only solve the problem but also understand the underlying principles of complex number arithmetic. This way, you'll be well-equipped to handle similar problems in the future. Remember, the key to mastering any mathematical concept is to practice and understand the 'why' behind the 'how'. So, let's get started and unlock the secrets of complex number multiplication!

What are Complex Numbers?

Before we jump into the multiplication, let's quickly recap what complex numbers are. A complex number is essentially a number that can be expressed in the form $a + bi$, where:

• $a$ is the real part.
• $b$ is the imaginary part.
• $i$ is the imaginary unit, defined as the square root of -1 (i.e., $i = \sqrt{-1}$).

Think of it like this: complex numbers extend the familiar number line we use for real numbers by adding another dimension – the imaginary dimension. This allows us to represent numbers that involve the square root of negative numbers, which are not possible within the realm of real numbers alone. This opens up a whole new world of mathematical possibilities and applications. Complex numbers aren't just abstract concepts; they play a crucial role in various fields, from electrical engineering and quantum mechanics to fluid dynamics and signal processing. So, understanding them is not just an academic exercise; it's a key to unlocking deeper insights in many areas of science and technology.

The Importance of i

The imaginary unit, i, is the heart and soul of complex numbers. It's what allows us to deal with the square roots of negative numbers. Remember, in the real number system, you can't take the square root of a negative number because no real number, when multiplied by itself, gives a negative result. But i changes the game! Since $i = \sqrt{-1}$, it follows that $i^2 = -1$. This seemingly simple definition has profound implications. It allows us to express the square root of any negative number in terms of i. For example, $\sqrt{-9}$ can be written as $\sqrt{9 \times -1} = \sqrt{9} \times \sqrt{-1} = 3i$. This is a fundamental concept that you'll use repeatedly when working with complex numbers. Understanding the properties of i, particularly that $i^2 = -1$, is absolutely essential for performing arithmetic operations like addition, subtraction, multiplication, and division with complex numbers. It's the key that unlocks the door to the world of complex number manipulation.

Examples of Complex Numbers

To solidify your understanding, let's look at a few examples of complex numbers:

• $3 + 2i$ (real part is 3, imaginary part is 2)
• $-1 - i$ (real part is -1, imaginary part is -1)
• $5i$ (real part is 0, imaginary part is 5 – this is a purely imaginary number)
• $7$ (real part is 7, imaginary part is 0 – this is a real number, but it can also be considered a complex number with a zero imaginary part)

These examples highlight the versatility of complex numbers. They can encompass both real numbers (when the imaginary part is zero) and purely imaginary numbers (when the real part is zero). The general form $a + bi$ elegantly captures this inclusivity. Visualizing complex numbers on a complex plane (with the real part on the x-axis and the imaginary part on the y-axis) can further enhance your understanding. It allows you to represent complex numbers as points or vectors, making it easier to grasp their geometric properties and relationships. As you encounter more examples and work through different problems, you'll become increasingly comfortable identifying and working with complex numbers in various forms.

Multiplying Complex Numbers: The FOIL Method

Now, let's get to the heart of the problem: multiplying $(6 - 5i)^2$. This means we're multiplying the complex number $(6 - 5i)$ by itself. The best way to approach this is by using the FOIL method, which stands for:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of each binomial.
• Inner: Multiply the inner terms of each binomial.
• Last: Multiply the last terms of each binomial.

This method is simply a systematic way of applying the distributive property, ensuring that you multiply each term in the first binomial by each term in the second binomial. It's a reliable technique that works for any binomial multiplication, not just complex numbers. When dealing with complex numbers, the FOIL method helps you break down the multiplication into manageable steps, making it less prone to errors. Remember, the key is to be organized and keep track of each term. Once you've mastered the FOIL method, you can confidently tackle more complex multiplications involving complex numbers and polynomials.

Applying FOIL to $(6 - 5i)^2$

Let's apply the FOIL method to our problem, $(6 - 5i)^2$ which is the same as $(6 - 5i)(6 - 5i)$:

• First: $6 * 6 = 36$
• Outer: $6 * (-5i) = -30i$
• Inner: $(-5i) * 6 = -30i$
• Last: $(-5i) * (-5i) = 25i^2$

So, after applying the FOIL method, we have: $36 - 30i - 30i + 25i^2$. This is a crucial intermediate step. It's important to meticulously perform each multiplication and keep track of the signs. A common mistake is to overlook the negative signs, especially when dealing with the imaginary terms. Double-checking your work at this stage can save you from errors later on. Remember, the FOIL method is just a tool to help you organize the multiplication process. The real magic happens when you simplify the expression by combining like terms and using the fact that $i^2 = -1$. So, let's move on to the simplification step and see how the imaginary unit transforms our result.

Simplifying the Expression

Now we have $36 - 30i - 30i + 25i^2$. The next step is to simplify this expression. This involves two main steps:

1. Combine like terms: We can combine the two imaginary terms, $-30i$ and $-30i$, to get $-60i$.
2. Substitute $i^2$ with -1: Remember that $i^2 = -1$. So, we can replace $25i^2$ with $25 * (-1) = -25$.

Combining like terms is a fundamental algebraic skill, and it's essential for simplifying expressions involving complex numbers. Identifying and grouping the real and imaginary terms correctly is key to arriving at the correct answer. The substitution of $i^2$ with -1 is the cornerstone of complex number simplification. It's this substitution that allows us to transform the expression into the standard form of a complex number, $a + bi$. Without this step, we'd be stuck with an expression that still contains $i^2$, which doesn't fit the standard form. So, remember to always look for opportunities to substitute $i^2$ with -1 when simplifying complex number expressions.

Putting it All Together

After combining like terms and substituting for $i^2$, our expression becomes:

$36 - 60i - 25$

Now, we can combine the real terms, 36 and -25, to get 11. This gives us our final answer:

$11 - 60i$

And there you have it! We've successfully multiplied $(6 - 5i)^2$ and arrived at the simplified complex number $11 - 60i$. This final step demonstrates the elegance of complex number arithmetic. By combining the real and imaginary terms, we express the result in the standard form $a + bi$, which is the conventional way of representing complex numbers. This form makes it easy to compare and perform further operations on complex numbers. The entire process, from applying the FOIL method to simplifying using $i^2 = -1$, highlights the importance of following a systematic approach and paying attention to detail. With practice, you'll become more fluent in these techniques and be able to confidently tackle a wide range of complex number problems.

The Answer

Therefore, the answer is B) $11 - 60i$. This matches our calculated result, confirming that we've correctly applied the FOIL method and simplified the expression. It's always a good practice to double-check your answer, especially in mathematics. This helps you identify any potential errors and reinforces your understanding of the concepts involved. In this case, we can be confident that our answer is correct, thanks to the careful step-by-step approach we've taken. Remember, accuracy and clarity are crucial in mathematics. Showing your work and double-checking your answers are hallmarks of a proficient mathematician. So, keep practicing, keep asking questions, and you'll continue to improve your skills in complex number arithmetic and beyond.

Practice Makes Perfect

To really master complex number multiplication, it's essential to practice! Try working through similar problems with different complex numbers. You can also explore other operations with complex numbers, such as addition, subtraction, and division. The more you practice, the more comfortable you'll become with the concepts and techniques involved. 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 keep pushing yourself to improve. There are numerous online resources and textbooks that offer practice problems and detailed explanations of complex number arithmetic. Take advantage of these resources and challenge yourself with increasingly complex problems. Remember, mathematics is a skill that improves with practice. So, keep at it, and you'll be amazed at how far you can go!

Further Exploration

If you're feeling adventurous, you can also delve deeper into the world of complex numbers. Explore topics like:

• The complex plane: A graphical representation of complex numbers.
• Polar form of complex numbers: Expressing complex numbers using magnitude and angle.
• De Moivre's Theorem: A powerful tool for raising complex numbers to powers.

These topics build upon the fundamental concepts we've covered today and open up a whole new realm of mathematical possibilities. The complex plane provides a visual way to understand complex numbers and their relationships. The polar form offers an alternative representation that is particularly useful for certain operations. De Moivre's Theorem is a powerful tool that simplifies the process of raising complex numbers to powers. Exploring these topics will not only deepen your understanding of complex numbers but also expose you to more advanced mathematical concepts and techniques. So, if you're curious and eager to learn more, don't hesitate to dive deeper into the fascinating world of complex numbers!

By understanding the basics and practicing regularly, you'll be well on your way to conquering complex numbers! Keep up the great work!