Multiply And Simplify (3-5i)(2+3i) A Step-by-Step Guide

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Hey everyone! Today, we're diving into the fascinating world of complex numbers. Specifically, we're going to tackle the problem of multiplying and simplifying the expression (3 - 5i)(2 + 3i). Don't worry if complex numbers seem a bit intimidating at first; we'll break it down step by step, and by the end of this guide, you'll be a pro at handling these types of calculations. So, grab your pencils and let's get started!

Understanding Complex Numbers

Before we jump into the multiplication, let's quickly recap what complex numbers are all about. At its core, a complex number is simply a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. This imaginary unit, i, is defined as the square root of -1. This might sound a little weird, but it opens up a whole new dimension in mathematics, allowing us to solve equations that have no solutions in the realm of real numbers alone.

The a part of a + bi is called the real part, and the b part is called the imaginary part. So, in the complex number 3 - 5i, the real part is 3, and the imaginary part is -5. Similarly, in 2 + 3i, the real part is 2, and the imaginary part is 3.

Complex numbers are used extensively in various fields, including electrical engineering, quantum mechanics, and fluid dynamics. They provide a powerful tool for modeling phenomena that involve oscillations, rotations, and wave behavior. Now that we've refreshed our understanding of complex numbers, let's move on to the multiplication process.

Multiplying Complex Numbers: The FOIL Method

When it comes to multiplying two complex numbers, we can use a method that's very familiar from algebra: the FOIL method. FOIL stands for First, Outer, Inner, Last, and it's a handy mnemonic for remembering how to multiply two binomials. In our case, (3 - 5i) and (2 + 3i) are both binomials, so FOIL is the perfect technique to use.

Let's break down the FOIL method in the context of our problem:

  1. First: Multiply the first terms of each binomial. In our case, that's 3 * 2, which equals 6.
  2. Outer: Multiply the outer terms of the binomials. That's 3 * 3i, which equals 9i.
  3. Inner: Multiply the inner terms of the binomials. That's -5i * 2, which equals -10i.
  4. Last: Multiply the last terms of each binomial. That's -5i * 3i, which equals -15i². Remember that i² = -1, this is a crucial point!.

So, after applying the FOIL method, we have:

(3 - 5i)(2 + 3i) = 6 + 9i - 10i - 15i²

Now, we need to simplify this expression. Let's move on to the next step.

Simplifying the Expression

Now that we've expanded the expression using the FOIL method, it's time to simplify it. This involves two main steps: combining like terms and dealing with the i² term.

Combining Like Terms

In our expression, 6 + 9i - 10i - 15i², we have two terms that involve i: 9i and -10i. These are like terms, and we can combine them by simply adding their coefficients. 9i - 10i equals -1i, which we can write as just -i.

So, our expression now looks like this:

6 - i - 15i²

Dealing with i²

This is where the magic of complex numbers really shines. Remember that i is the imaginary unit, defined as the square root of -1. This means that i² is equal to -1. We can substitute -1 for i² in our expression:

6 - i - 15(-1)

Now, we can simplify further by multiplying -15 by -1, which gives us +15:

6 - i + 15

Final Simplification

We're almost there! Now we just need to combine the real terms, 6 and 15, which add up to 21. This gives us our final simplified expression:

21 - i

The Final Answer

So, after multiplying and simplifying (3 - 5i)(2 + 3i), we arrive at the complex number 21 - i. This is our final answer, expressed in the standard form a + bi, where a is 21 and b is -1.

Key Takeaways and Practice Problems

Guys, let's quickly recap the key steps we've covered in this guide:

  1. Understand Complex Numbers: Remember that a complex number is in the form a + bi, where i is the imaginary unit (√-1) and i² = -1.
  2. Use the FOIL Method: When multiplying two complex numbers, use the FOIL method (First, Outer, Inner, Last) to expand the expression.
  3. Combine Like Terms: Combine the real terms and the imaginary terms (terms with i) separately.
  4. Substitute i² = -1: Replace any instances of i² with -1.
  5. Simplify and Express in Standard Form: Simplify the expression and write the final answer in the form a + bi.

To solidify your understanding, let's try a few practice problems:

  1. Multiply and simplify (2 + i)(4 - 3i).
  2. Multiply and simplify (-1 + 2i)(3 + i).
  3. Multiply and simplify (5 - 2i)(5 + 2i).

Working through these problems will help you become more comfortable with multiplying and simplifying complex numbers. Remember, practice makes perfect!

Common Mistakes to Avoid

When working with complex numbers, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you get the correct answer.

  1. Forgetting that i² = -1: This is the most crucial point to remember! Failing to substitute -1 for i² will lead to an incorrect answer.
  2. Incorrectly Applying the FOIL Method: Make sure you multiply each term in the first binomial by each term in the second binomial. Double-check your multiplications to avoid errors.
  3. Combining Unlike Terms: Only combine real terms with real terms and imaginary terms with imaginary terms. Don't try to add a real number to an imaginary number.
  4. Sign Errors: Pay close attention to the signs (positive and negative) when multiplying and combining terms. A small sign error can throw off your entire calculation.

By keeping these common mistakes in mind, you'll be well on your way to mastering complex number multiplication.

Real-World Applications of Complex Numbers

Okay, so we've learned how to multiply and simplify complex numbers, but you might be wondering,