In Exercises 65–70, perform the indicated operation(s) and write the result in standard form. (4 - i)² - (1 + 2i)²

Guida verificata passo dopo passo

Step 1: Recall the formula for squaring a binomial: $(a + b)^2 = a^2 + 2ab + b^2$. Apply this formula to $(4 - i)^2$, where $a = 4$ and $b = -i$. Expand it as $4^2 + 2(4)(-i) + (-i)^2$.

Step 2: Similarly, apply the same formula to $(1 + 2i)^2$, where $a = 1$ and $b = 2i$. Expand it as $1^2 + 2(1)(2i) + (2i)^2$.

Step 3: Simplify each term in the expansions. For $(4 - i)^2$, calculate $4^2 = 16$, $2(4)(-i) = -8i$, and $(-i)^2 = -1$. Combine these to get $16 - 8i - 1$. For $(1 + 2i)^2$, calculate $1^2 = 1$, $2(1)(2i) = 4i$, and $(2i)^2 = -4$. Combine these to get $1 + 4i - 4$.

Step 4: Subtract the result of $(1 + 2i)^2$ from the result of $(4 - i)^2$. Write this as $(16 - 8i - 1) - (1 + 4i - 4)$. Distribute the negative sign across the second set of parentheses.

Step 5: Combine like terms (real parts and imaginary parts) from the subtraction in Step 4. Simplify the expression to write the result in standard form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding how to manipulate complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Video consigliato:

04:22

Dividing Complex Numbers

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. When performing operations on complex numbers, the result should be expressed in this form to clearly identify the real and imaginary components. This is particularly important in problems involving addition, subtraction, or multiplication of complex numbers.

Video consigliato:

05:02

Multiplying Complex Numbers

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, such as (a + b)^n. The formula for binomial expansion involves using the binomial coefficients, which can be found in Pascal's triangle. In the context of complex numbers, this concept is crucial for squaring binomials like (4 - i)^2 and (1 + 2i)^2 to simplify the expression before combining like terms.

Video consigliato: