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$}$

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Introduction

In mathematics, equations involving complex numbers are a fundamental concept in algebra and analysis. These equations often involve the use of imaginary numbers, which are a crucial part of solving problems in various fields, including physics, engineering, and computer science. In this article, we will explore the values of $a$ and $b$ that make the given equation true, involving complex numbers and their properties.

Understanding Complex Numbers

Complex numbers are numbers 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 denoted by $a$, and the imaginary part is denoted by $b$. Complex numbers can be added, subtracted, multiplied, and divided, just like real numbers, but with some additional rules and properties.

The Given Equation

The given equation is:

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

To solve this equation, we need to simplify the expressions inside the parentheses and then combine like terms.

Simplifying the Expressions

Let's start by simplifying the expression inside the first set of parentheses:

$4+\sqrt{-49}$

We can rewrite $\sqrt{-49}$ as $\sqrt{(-7)^2}$, which is equal to $7i$. Therefore, the expression becomes:

$4+7i$

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

$\sqrt{(-4)^2}+\sqrt{-324}$

We can rewrite $\sqrt{(-4)^2}$ as $4$, and $\sqrt{-324}$ as $\sqrt{(-18)^2}$, which is equal to $18i$. Therefore, the expression becomes:

$4+18i$

Substituting the Simplified Expressions

Now that we have simplified the expressions, we can substitute them back into the original equation:

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

Expanding and Combining Like Terms

Next, we need to expand and combine like terms:

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

Combining like terms, we get:

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

Equating Real and Imaginary Parts

Since the equation is equal to $a+bi$, we can equate the real and imaginary parts separately:

$-4=a$

$-29=b$

Conclusion

In conclusion, the values of $a$ and $b$ that make the given equation true are $a=-4$ and $b=-29$. These values satisfy the equation and demonstrate the properties of complex numbers.

Final Answer

The final answer is:

$a = \boxed{-4}$

$b = \boxed{-29}$

Discussion

This problem involves the use of complex numbers and their properties. The equation is solved by simplifying the expressions inside the parentheses and then combining like terms. The real and imaginary parts are equated separately to find the values of $a$ and $b$. This problem demonstrates the importance of understanding complex numbers and their properties in mathematics.

Related Topics

• Complex numbers
• Imaginary numbers
• Algebra
• Analysis
• Physics
• Engineering
• Computer science

References

• [1] "Complex Numbers" by Math Is Fun
• [2] "Imaginary Numbers" by Khan Academy
• [3] "Algebra" by MIT OpenCourseWare
• [4] "Analysis" by University of Michigan
• [5] "Physics" by OpenStax
• [6] "Engineering" by Coursera
• [7] "Computer Science" by edX
  

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Introduction

In our previous article, we explored the values of $a$ and $b$ that make the given equation true, involving complex numbers and their properties. In this article, we will provide a Q&A section to further clarify any doubts and provide additional information on the topic.

Q&A

Q: What is the difference between real and imaginary numbers?

A: Real numbers are numbers that can be expressed without any imaginary part, such as 3, 4, or -5. Imaginary numbers, on the other hand, are numbers that can be expressed with an imaginary part, such as 3i, 4i, or -5i.

Q: What is the value of i?

A: The value of i is the imaginary unit, which satisfies the equation i^2 = -1. This means that i is equal to the square root of -1.

Q: How do you add and subtract complex numbers?

A: To add complex numbers, you add the real parts and the imaginary parts separately. For example, (3+4i) + (2+5i) = (3+2) + (4i+5i) = 5 + 9i. To subtract complex numbers, you subtract the real parts and the imaginary parts separately. For example, (3+4i) - (2+5i) = (3-2) + (4i-5i) = 1 - i.

Q: How do you multiply complex numbers?

A: To multiply complex numbers, you multiply the real parts and the imaginary parts separately, and then combine the results. For example, (3+4i) * (2+5i) = (32 + 35i + 4i2 + 4i5i) = (6 + 15i + 8i - 20) = -14 + 23i.

Q: How do you divide complex numbers?

A: To divide complex numbers, you multiply the numerator and the denominator by the conjugate of the denominator. For example, (3+4i) / (2+5i) = ((3+4i) * (2-5i)) / ((2+5i) * (2-5i)) = ((6 - 15i + 8i - 20)) / ((4 + 10i - 10i - 25)) = (-14 + 8i) / (-21) = (14/21) - (8/21)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 of the imaginary part. For example, the conjugate of 3+4i is 3-4i.

Q: What is the modulus of a complex number?

A: The modulus of a complex number is the distance from the origin to the point representing the complex number in the complex plane. It is calculated as the square root of the sum of the squares of the real and imaginary parts. For example, the modulus of 3+4i is sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.

Q: What is the argument of a complex number?

A: The argument of a complex number is the angle between the positive real axis and the line segment connecting the origin to the point representing the complex number in the complex plane. It is calculated as the inverse tangent of the imaginary part divided by the real part. For example, the argument of 3+4i is arctan(4/3).

Conclusion

In conclusion, the values of $a$ and $b$ that make the given equation true are $a=-4$ and $b=-29$. We have also provided a Q&A section to further clarify any doubts and provide additional information on the topic. We hope this article has been helpful in understanding complex numbers and their properties.

Final Answer

The final answer is:

$a = \boxed{-4}$

$b = \boxed{-29}$

Discussion

This problem involves the use of complex numbers and their properties. The equation is solved by simplifying the expressions inside the parentheses and then combining like terms. The real and imaginary parts are equated separately to find the values of $a$ and $b$. This problem demonstrates the importance of understanding complex numbers and their properties in mathematics.

Related Topics

• Complex numbers
• Imaginary numbers
• Algebra
• Analysis
• Physics
• Engineering
• Computer science

References

• [1] "Complex Numbers" by Math Is Fun
• [2] "Imaginary Numbers" by Khan Academy
• [3] "Algebra" by MIT OpenCourseWare
• [4] "Analysis" by University of Michigan
• [5] "Physics" by OpenStax
• [6] "Engineering" by Coursera
• [7] "Computer Science" by edX