/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 42 Perform the operation and write ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 42

Perform the operation and write the result in standard form. $$(1-2 i)^{2}-(1+2 i)^{2}$$

Short Answer

Expert verified
The result of the operation is \( -8i \).

Step by step solution

01

Squaring the first complex number

Start by performing the operation \( (1-2i)^{2} \). Use the square of binomial formula \( (a - b)^{2} = a^{2} - 2ab + b^{2} \). Here, \( a=1 \) and \( b=2i \). So, \( (1-2i)^2 = 1^{2} - 2*1*2i + (2i)^2 = 1 - 4i + 4i^2 = 1 - 4i - 4 \) because \( i^{2} = -1 \). Hence, \( (1-2i)^{2} = 1 - 4i - 4 = -3 - 4i \).
02

Squaring the second complex number

Now, perform the operation \( (1+2i)^{2} \). Use the square of binomial formula \( (a + b)^{2} = a^{2} + 2ab + b^{2} \). Here, \( a=1 \) and \( b=2i \). So, \( (1+2i)^2 = 1^{2} + 2*1*2i + (2i)^2 = 1 + 4i + 4i^{2} = 1 + 4i - 4 \) because \( i^{2} = -1 \). Hence, \( (1+2i)^2 = 1 + 4i - 4 = -3 + 4i \).
03

Subtracting the second squared complex number from the first

Now, perform the subtraction: \( (1-2i)^{2} - (1+2i)^{2} = (-3 - 4i) - (-3 + 4i) \). Verify that only the imaginary parts of the complex numbers cancel out, and the result will be: \( -4i - 4i = -8i \).
04

Write the result in standard form

Given that the standard form of a complex number is \( a+bi \), where \( a \) and \( b \) are real numbers, and \( b \) is the coefficient of \( i \), the result in standard form is simply \( 0 - 8i \). However, usually when the real part is zero we omit it, so the final result is \( -8i \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{3}{2} x^{2}-\frac{23}{2} x+6=\frac{1}{2}\left(2 x^{3}-3 x^{2}-23 x+12\right)$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=x^{2}+10 x+17$$

Simplify the complex number and write it in standard form. $$(-i)^{6}$$

Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) $$2,5+i$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(z)=z^{2}-2 z+2$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.