What Is The Product Of { -3+6i$}$ And Its Conjugate?A. 27 B. 45 C. ${ 45-36i\$} D. ${ 27-36i\$}

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Introduction

In mathematics, the conjugate of a complex number is another complex number that has the same real part but an opposite imaginary part. The product of a complex number and its conjugate is a real number, which can be calculated using the formula: (a + bi)(a - bi) = a^2 + b^2. In this article, we will explore the product of {-3+6i$}$ and its conjugate.

What is a Complex Number?

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i^2 = -1. The real part of a complex number is the part that is not multiplied by i, and the imaginary part is the part that is multiplied by i.

What is the Conjugate of a Complex Number?

The conjugate of a complex number a + bi is a - bi. For example, the conjugate of 3 + 4i is 3 - 4i. The conjugate of a complex number has the same real part but an opposite imaginary part.

Calculating the Product of a Complex Number and Its Conjugate

To calculate the product of a complex number and its conjugate, we can use the formula: (a + bi)(a - bi) = a^2 + b^2. This formula is derived from the fact that (a + bi)(a - bi) = a^2 - abi + abi - b2i2 = a^2 + b^2.

Calculating the Product of {-3+6i$}$ and Its Conjugate

To calculate the product of {-3+6i$}$ and its conjugate, we need to find the conjugate of {-3+6i$}$. The conjugate of {-3+6i$}$ is {-3-6i$}$. Now, we can calculate the product using the formula: (a + bi)(a - bi) = a^2 + b^2.

Step 1: Identify the Real and Imaginary Parts

The real part of {-3+6i$}$ is -3, and the imaginary part is 6.

Step 2: Calculate the Product

Using the formula (a + bi)(a - bi) = a^2 + b^2, we can calculate the product as follows:

(-3)^2 + (6)^2 = 9 + 36 = 45

Step 3: Write the Product in the Form a + bi

Since the product is a real number, we can write it in the form a + bi, where a is the product and b is 0. Therefore, the product of {-3+6i$}$ and its conjugate is 45 + 0i, which can be written as ${45\$}.

Conclusion

In this article, we calculated the product of {-3+6i$}$ and its conjugate using the formula (a + bi)(a - bi) = a^2 + b^2. We found that the product is 45, which can be written as ${45\$}. This result is consistent with the answer choices provided.

Answer

The correct answer is C. ${45\$}.

Frequently Asked Questions

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number a + bi is a - bi.

Q: How do I calculate the product of a complex number and its conjugate?

A: To calculate the product of a complex number and its conjugate, you can use the formula: (a + bi)(a - bi) = a^2 + b^2.

Q: What is the product of {-3+6i$}$ and its conjugate?

A: The product of {-3+6i$}$ and its conjugate is 45.

References

Introduction

In our previous article, we explored the product of a complex number and its conjugate. In this article, we will answer some frequently asked questions about complex numbers and conjugates.

Q&A

Q: What is a complex number?

A: A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i^2 = -1.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number a + bi is a - bi.

Q: How do I calculate the product of a complex number and its conjugate?

A: To calculate the product of a complex number and its conjugate, you can use the formula: (a + bi)(a - bi) = a^2 + b^2.

Q: What is the product of {-3+6i$}$ and its conjugate?

A: The product of {-3+6i$}$ and its conjugate is 45.

Q: Can I use the conjugate of a complex number to simplify expressions?

A: Yes, you can use the conjugate of a complex number to simplify expressions. For example, if you have the expression (a + bi)(a - bi), you can simplify it to a^2 + b^2 using the formula.

Q: How do I find the conjugate of a complex number in the form a + bi?

A: To find the conjugate of a complex number in the form a + bi, you can simply change the sign of the imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.

Q: Can I use the conjugate of a complex number to solve equations?

A: Yes, you can use the conjugate of a complex number to solve equations. For example, if you have the equation (a + bi) = c, you can multiply both sides by the conjugate of (a + bi) to eliminate the imaginary part.

Q: What is the difference between a complex number and a real number?

A: A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. A real number is a number that can be expressed in the form a, where a is a real number.

Q: Can I add or subtract complex numbers?

A: Yes, you can add or subtract complex numbers. For example, if you have the complex numbers 3 + 4i and 2 - 3i, you can add them to get 5 + i.

Q: Can I multiply complex numbers?

A: Yes, you can multiply complex numbers. For example, if you have the complex numbers 3 + 4i and 2 - 3i, you can multiply them to get -9 + 5i.

Conclusion

In this article, we answered some frequently asked questions about complex numbers and conjugates. We hope that this article has helped you to understand complex numbers and conjugates better.

Frequently Asked Questions

Q: What is the difference between a complex number and a real number?

A: A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. A real number is a number that can be expressed in the form a, where a is a real number.

Q: Can I use the conjugate of a complex number to simplify expressions?

A: Yes, you can use the conjugate of a complex number to simplify expressions. For example, if you have the expression (a + bi)(a - bi), you can simplify it to a^2 + b^2 using the formula.

Q: How do I find the conjugate of a complex number in the form a + bi?

A: To find the conjugate of a complex number in the form a + bi, you can simply change the sign of the imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.

References