4. Which Expression Is Equivalent To \[$(2x-i)^2 - (2x-i)(2x+3i)\$\], Where \[$i\$\] Is The Imaginary Unit And \[$x\$\] Is A Real Number?A. \[$-4 - 8xi\$\]B. \[$-4 - 4xi\$\]C. 2D. \[$8x - 4i\$\]
Introduction
In mathematics, complex expressions can be daunting to simplify, especially when dealing with imaginary units and variables. However, with a clear understanding of the rules and techniques involved, simplifying complex expressions becomes a manageable task. In this article, we will focus on simplifying the expression {(2x-i)^2 - (2x-i)(2x+3i)$}$, where {i$}$ is the imaginary unit and {x$}$ is a real number.
Understanding the Imaginary Unit
Before we dive into simplifying the expression, it's essential to understand the concept of the imaginary unit. The imaginary unit, denoted by {i$}$, is defined as the square root of -1. This means that {i^2 = -1$}$. The imaginary unit is used to extend the real number system to the complex number system, which includes all numbers of the form {a + bi$}$, where {a$}$ and {b$}$ are real numbers and {i$}$ is the imaginary unit.
Simplifying the Expression
To simplify the expression {(2x-i)^2 - (2x-i)(2x+3i)$}$, we will use the distributive property and the rules of exponents. We will start by expanding the first term, {(2x-i)^2$}$.
{(2x-i)^2 = (2x)^2 - 2(2x)(i) + i^2$}$
Using the rules of exponents, we can simplify the expression as follows:
{(2x)^2 = 4x^2$}$
{-2(2x)(i) = -4xi$}$
{i^2 = -1$}$
Substituting these values back into the expression, we get:
{(2x-i)^2 = 4x^2 - 4xi - 1$}$
Next, we will expand the second term, {(2x-i)(2x+3i)$}$.
{(2x-i)(2x+3i) = (2x)(2x) + (2x)(3i) - i(2x) - i(3i)$}$
Using the distributive property, we can simplify the expression as follows:
{(2x)(2x) = 4x^2$}$
{(2x)(3i) = 6xi$}$
{-i(2x) = -2xi$}$
{-i(3i) = 3$}$
Substituting these values back into the expression, we get:
{(2x-i)(2x+3i) = 4x^2 + 6xi - 2xi + 3$}$
Combining like terms, we get:
{(2x-i)(2x+3i) = 4x^2 + 4xi + 3$}$
Now, we can substitute the simplified expressions back into the original expression:
{(2x-i)^2 - (2x-i)(2x+3i) = (4x^2 - 4xi - 1) - (4x^2 + 4xi + 3)$}$
Using the distributive property, we can simplify the expression as follows:
{(4x^2 - 4xi - 1) - (4x^2 + 4xi + 3) = 4x^2 - 4xi - 1 - 4x^2 - 4xi - 3$}$
Combining like terms, we get:
${4x^2 - 4xi - 1 - 4x^2 - 4xi - 3 = -4 - 8xi\$}
Therefore, the simplified expression is {-4 - 8xi$}$.
Conclusion
In this article, we simplified the complex expression {(2x-i)^2 - (2x-i)(2x+3i)$}$ using the distributive property and the rules of exponents. We started by expanding the first term and then expanded the second term. Finally, we combined like terms to get the simplified expression {-4 - 8xi$}$. This expression is equivalent to the original expression, and it demonstrates the importance of understanding the rules and techniques involved in simplifying complex expressions.
Answer
The correct answer is:
- A. {-4 - 8xi$}$
Introduction
In our previous article, we simplified the complex expression {(2x-i)^2 - (2x-i)(2x+3i)$}$ using the distributive property and the rules of exponents. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in simplifying complex expressions.
Q: What is the imaginary unit?
A: The imaginary unit, denoted by {i$}$, is defined as the square root of -1. This means that {i^2 = -1$}$. The imaginary unit is used to extend the real number system to the complex number system, which includes all numbers of the form {a + bi$}$, where {a$}$ and {b$}$ are real numbers and {i$}$ is the imaginary unit.
Q: How do I simplify a complex expression?
A: To simplify a complex expression, you can use the distributive property and the rules of exponents. Start by expanding the first term, and then expand the second term. Finally, combine like terms to get the simplified expression.
Q: What is the distributive property?
A: The distributive property is a rule that allows you to multiply a single term by multiple terms. For example, {a(b + c) = ab + ac$}$. This property is used to expand expressions and simplify complex expressions.
Q: What are the rules of exponents?
A: The rules of exponents are a set of rules that govern the behavior of exponents in mathematical expressions. For example, {a^m \cdot a^n = a^{m+n}$}$ and {(am)n = a^{mn}$}$. These rules are used to simplify complex expressions and solve equations.
Q: How do I combine like terms?
A: To combine like terms, you need to identify the terms that have the same variable and exponent. For example, ${2x + 3x = 5x\$}. Combine the coefficients of the like terms to get the simplified expression.
Q: What is the difference between a real number and a complex number?
A: A real number is a number that can be expressed as a single value, such as ${3\$} or {-4$}$. A complex number, on the other hand, is a number that can be expressed as a sum of a real number and an imaginary number, such as ${3 + 4i\$} or {-2 - 3i$}$.
Q: How do I simplify a complex fraction?
A: To simplify a complex fraction, you need to multiply the numerator and denominator by the conjugate of the denominator. For example, {\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$}$. Simplify the expression to get the final answer.
Q: What are some common mistakes to avoid when simplifying complex expressions?
A: Some common mistakes to avoid when simplifying complex expressions include:
- Not using the distributive property correctly
- Not following the rules of exponents
- Not combining like terms correctly
- Not simplifying the expression fully
Conclusion
In this article, we provided a Q&A guide to help you understand the concepts and techniques involved in simplifying complex expressions. We covered topics such as the imaginary unit, the distributive property, the rules of exponents, and combining like terms. We also discussed common mistakes to avoid when simplifying complex expressions. By following these guidelines, you can simplify complex expressions with confidence.
Additional Resources
For more information on simplifying complex expressions, check out the following resources:
- Khan Academy: Simplifying Complex Expressions
- Mathway: Simplifying Complex Expressions
- Wolfram Alpha: Simplifying Complex Expressions
Practice Problems
Try these practice problems to test your skills in simplifying complex expressions:
- Simplify the expression {(3x + 2i)^2 - (3x + 2i)(4x - 5i)$}$
- Simplify the expression {\frac{2x + 3i}{x - 2i}$}$
- Simplify the expression {(x + 4i)^2 + (x + 4i)(3x - 2i)$}$
Answer Key
- Simplify the expression {(3x + 2i)^2 - (3x + 2i)(4x - 5i)$]: [-13x^2 + 14xi - 9\$}
- Simplify the expression {\frac2x + 3i}{x - 2i}$]{x - 2i} \cdot \frac{x + 2i}{x + 2i} = \frac{(2x + 3i)(x + 2i)}{x^2 + 4}$}$
- Simplify the expression {(x + 4i)^2 + (x + 4i)(3x - 2i)$]: [-7x^2 + 10xi + 16\$}