Type The Correct Answer In Each Box.What Values Of $a$ And $b$ Make This Equation True?${ (4+\sqrt{-49})-2\left(\sqrt{(-4)^2}+\sqrt{-324}\right)=a+b I }$ { a=\square \} ${ b=\square }$

Introduction

In this article, we will explore the concept of complex numbers and how to solve equations involving them. Complex numbers are numbers that have both real and imaginary parts. They are used to represent quantities that have both magnitude and direction. In this case, we will be solving an equation that involves complex numbers and finding the values of $a$ and $b$ that make the equation true.

Understanding Complex Numbers

Complex numbers are numbers that can be written 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 $i$.

The Equation

The equation we will be solving is:

$(4+\sqrt{-49})-2\left(\sqrt{(-4)^2}+\sqrt{-324}\right)=a+b i$

To solve this equation, we need to simplify the left-hand side and then equate the real and imaginary parts.

Simplifying the Left-Hand Side

First, let's simplify the expression inside the parentheses:

$\sqrt{-49} = \sqrt{-1 \cdot 49} = \sqrt{-1} \cdot \sqrt{49} = 7i$

Next, let's simplify the expression inside the second set of parentheses:

$\sqrt{(-4)^2} = \sqrt{16} = 4$

$\sqrt{-324} = \sqrt{-1 \cdot 324} = \sqrt{-1} \cdot \sqrt{324} = 18i$

Now, let's substitute these simplified expressions back into the original equation:

$(4+7i)-2(4+18i)=a+b i$

Distributing the 2

Next, let's distribute the 2 to the terms inside the parentheses:

$4+7i-8-36i=a+b i$

Combining Like Terms

Now, let's combine like terms:

$-4-29i=a+b i$

Equating Real and Imaginary Parts

Since the left-hand side of the equation is a complex number, we can equate the real and imaginary parts separately:

$-4=a$

$-29=b$

Conclusion

In this article, we solved an equation involving complex numbers and found the values of $a$ and $b$ that make the equation true. We simplified the left-hand side of the equation, distributed the 2, combined like terms, and equated the real and imaginary parts. The final values of $a$ and $b$ are $-4$ and $-29$, respectively.

Final Answer

The final answer is:

$a=-4$

$b=-29$

Discussion

This problem is a great example of how to solve equations involving complex numbers. It requires a good understanding of complex numbers and how to simplify expressions involving them. The key to solving this problem is to simplify the left-hand side of the equation and then equate the real and imaginary parts.

Related Problems

If you want to practice solving equations involving complex numbers, here are some related problems:

• Solve the equation $(3+2i)-4(1+3i)=a+b i$
• Solve the equation $(2-3i)+2(4-2i)=a+b i$
• Solve the equation $(5+4i)-3(2-3i)=a+b i$

Introduction

In this article, we will answer some frequently asked questions about complex numbers. Complex numbers are numbers that have both real and imaginary parts. They are used to represent quantities that have both magnitude and direction. We will cover topics such as what complex numbers are, how to add and subtract complex numbers, and how to multiply and divide complex numbers.

Q: What are complex numbers?

A: Complex numbers are numbers that have both real and imaginary parts. They are used to represent quantities that have both magnitude and direction. A complex number is written 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: How do I add complex numbers?

A: To add complex numbers, you simply add the real parts and the imaginary parts separately. For example, if you have the complex numbers $3 + 4i$ and $2 + 5i$, you would add them as follows:

$(3 + 4i) + (2 + 5i) = (3 + 2) + (4i + 5i) = 5 + 9i$

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you simply subtract the real parts and the imaginary parts separately. For example, if you have the complex numbers $3 + 4i$ and $2 + 5i$, you would subtract them as follows:

$(3 + 4i) - (2 + 5i) = (3 - 2) + (4i - 5i) = 1 - i$

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you use the distributive property and the fact that $i^2 = -1$. For example, if you have the complex numbers $3 + 4i$ and $2 + 5i$, you would multiply them as follows:

$(3 + 4i)(2 + 5i) = (3)(2) + (3)(5i) + (4i)(2) + (4i)(5i)$

$= 6 + 15i + 8i + 20i^2$

$= 6 + 23i - 20$

$= -14 + 23i$

Q: How do I divide complex numbers?

A: To divide complex numbers, you multiply the numerator and denominator by the conjugate of the denominator. For example, if you have the complex numbers $3 + 4i$ and $2 + 5i$, you would divide them as follows:

$\frac{3 + 4i}{2 + 5i} = \frac{(3 + 4i)(2 - 5i)}{(2 + 5i)(2 - 5i)}$

$= \frac{6 - 15i + 8i - 20i^2}{4 - 25i^2}$

$= \frac{6 - 7i + 20}{4 + 25}$

$= \frac{26 - 7i}{29}$

$= \frac{26}{29} - \frac{7}{29}i$

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is the complex number with the opposite sign in the imaginary part. For example, the conjugate of $3 + 4i$ is $3 - 4i$.

Q: What is the absolute value of a complex number?

A: The absolute value of a complex number is the distance from the origin to the point in the complex plane. It is calculated using the formula $|a + bi| = \sqrt{a^2 + b^2}$.

Conclusion

In this article, we answered some frequently asked questions about complex numbers. We covered topics such as what complex numbers are, how to add and subtract complex numbers, and how to multiply and divide complex numbers. We also discussed the conjugate and absolute value of complex numbers. We hope this article has been helpful in understanding complex numbers.