Solving Complex Numbers: Find P And Q For (3-7i)(p+qi)i = 58i

Hey guys! Let's dive into a fun complex number problem today. We're going to figure out which values of p and q make the expression (3 - 7i)(p + qi)i equal to 58i. This might seem a bit daunting at first, but don't worry, we'll break it down step by step. The beauty of complex numbers lies in their predictable behavior, and by understanding the basic rules of complex number arithmetic, we can solve this problem quite easily. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we fully grasp what the problem is asking. We have the expression (3 - 7i)(p + qi)i, where p and q are real numbers we need to find. The i here represents the imaginary unit, which is defined as the square root of -1. Remember that i squared (i²) is equal to -1, a crucial property we'll use later. Our goal is to find the values of p and q that make this entire expression equal to 58i. This means the real part of the expression should be 0, and the imaginary part should be 58. This is a classic complex number problem that tests our understanding of complex number multiplication and equality. By working through this, we'll reinforce our knowledge of complex number arithmetic and problem-solving strategies. The expression involves multiplying two complex numbers (3 - 7i) and (p + qi), and then multiplying the result by i. Finally, we need to equate the result to 58i. To solve this, we will expand the expression, simplify it by using the property i² = -1, and then equate the real and imaginary parts to find the values of p and q.

Step-by-Step Solution

Okay, let's get our hands dirty with the math! Here’s how we can solve this problem:

1. Expand the Expression

First, we need to expand the expression (3 - 7i)(p + qi)i. Let’s start by multiplying the first two complex numbers, (3 - 7i) and (p + qi), using the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last):

(3 - 7i)(p + qi) = 3p + 3qi - 7pi - 7qi²

2. Simplify Using i² = -1

Remember that i² = -1. Let's substitute that into our expression:

3p + 3qi - 7pi - 7q(-1) = 3p + 3qi - 7pi + 7*q

Now, let's group the real and imaginary terms:

(3p + 7q) + (3q - 7p)i

3. Multiply by i

Next, we need to multiply the entire result by i:

[(3p + 7q) + (3q - 7p)i]i = (3p + 7q)i + (3q - 7p)i²

Again, we use i² = -1 to simplify:

(3p + 7q)i + (3q - 7p)(-1) = (3p + 7q)i - (3q - 7p)

Rearranging to group real and imaginary parts, we get:

(-3q + 7p) + (3p + 7q)i

4. Equate Real and Imaginary Parts

We know that the result should be equal to 58i. This means the real part must be 0, and the imaginary part must be 58. So, we can set up two equations:

Real part: -3q + 7p = 0

Imaginary part: 3p + 7q = 58

5. Solve the System of Equations

Now we have a system of two linear equations with two variables. We can solve this using substitution or elimination. Let’s use elimination. First, we can rearrange the first equation to express p in terms of q:

7p = 3q

p = (3/7)q

Now, substitute this expression for p into the second equation:

3*(3/7)q + 7q = 58

(9/7)q + 7q = 58

To get rid of the fraction, multiply the entire equation by 7:

9q + 49q = 406

58q = 406

Now, solve for q:

q = 406 / 58

q = 7

Great! Now that we have q, we can substitute it back into our expression for p:

p = (3/7)*7

p = 3

6. The Solution

So, we’ve found that p = 3 and q = 7. Let's double-check our answer by plugging these values back into the original expression:

(3 - 7i)(3 + 7i)i = (9 + 21i - 21i - 49i²)i = (9 + 49)i = 58i

Yep, it works!

Common Mistakes to Avoid

When working with complex numbers, there are a few common pitfalls to watch out for:

• Forgetting that i² = -1: This is the cornerstone of complex number simplification. Always remember to substitute -1 whenever you see i². Failing to do so will lead to incorrect answers.
• Incorrectly Distributing: When multiplying complex numbers, make sure you distribute each term properly (using FOIL or a similar method). Missing a term can throw off your entire calculation.
• Mixing Real and Imaginary Parts: Always keep the real and imaginary parts separate during calculations. Don't try to combine them until the very end when you're presenting your final answer.
• Sign Errors: Be extra careful with signs, especially when dealing with negative numbers and the imaginary unit. A small sign error can lead to a completely different answer.
• Not Checking Your Answer: It's always a good idea to plug your solution back into the original equation to make sure it works. This can help you catch any mistakes you might have made along the way.

By being mindful of these common mistakes, you can increase your accuracy and confidence when solving complex number problems. Practice is key, so don't hesitate to work through more examples and challenge yourself with different types of problems.

Practice Makes Perfect

To really master complex numbers, the key is practice, practice, practice! Try working through similar problems, and don't be afraid to change up the values or the form of the expression. The more you practice, the more comfortable you'll become with the rules and techniques involved. You can find plenty of practice problems in textbooks, online resources, or even create your own. Challenge yourself to solve problems in different ways, and always double-check your answers. Remember, the goal is not just to get the right answer, but also to understand the underlying concepts and develop your problem-solving skills. So, keep practicing, and you'll become a complex number whiz in no time!

Conclusion

So, the values of p and q that make the expression (3 - 7i)(p + qi)i equal to 58i are p = 3 and q = 7. We did it! This problem showcases how we can manipulate complex numbers using basic algebraic principles and the key property that i² = -1. By carefully expanding, simplifying, and equating real and imaginary parts, we can solve for unknown variables. Remember, guys, complex numbers might seem complex (pun intended!), but with a systematic approach and a bit of practice, you can conquer them. Keep practicing, and you'll be solving complex number problems like a pro in no time! This journey through complex numbers highlights the importance of understanding fundamental mathematical principles and applying them methodically to solve problems. As we continue to explore more advanced topics in mathematics, the skills and techniques we've learned here will serve as a solid foundation for our future endeavors. So, let's keep pushing the boundaries of our knowledge and embrace the beauty and complexity of mathematics.