Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 21

Chapter 2, Problem 21

In Exercises 21–28, divide and express the result in standard form. 2/(3 - i)

Verified step by step guidance

Step 1: Recognize that the denominator contains a complex number (3 - i). To simplify the expression, we need to eliminate the imaginary part from the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.

Step 2: The conjugate of (3 - i) is (3 + i). Multiply both the numerator and denominator of the fraction by (3 + i): $ \frac{2}{3 - i} \cdot \frac{3 + i}{3 + i} $.

Step 3: Expand the numerator by distributing 2 across (3 + i): $ 2(3 + i) = 6 + 2i $. The numerator becomes $ 6 + 2i $.

Step 4: Expand the denominator using the difference of squares formula: $ (3 - i)(3 + i) = 3^2 - i^2 $. Simplify further: $ 9 - (-1) = 9 + 1 = 10 $. The denominator becomes 10.

Step 5: Combine the results to form the simplified fraction: $ \frac{6 + 2i}{10} $. Finally, separate the real and imaginary parts by dividing each term in the numerator by 10: $ \frac{6}{10} + \frac{2i}{10} $. Simplify the fractions to express the result in standard form.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Division of Complex Numbers

Dividing complex numbers involves multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. This process eliminates the imaginary part in the denominator, allowing the result to be expressed in standard form, which is a + bi.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' and 'b' are real numbers. In this form, 'a' represents the real part and 'b' represents the imaginary part. Converting a complex number into standard form is crucial for clarity and further mathematical operations, making it easier to interpret and use in calculations.

Recommended video:

05:02

Multiplying Complex Numbers

Related Practice

Textbook Question

In Exercises 9–20, find each product and write the result in standard form. (5 - 2i)²

108

views

Textbook Question

In Exercises 15–26, use graphs to find each set. (- ∞, 5) ⋃ [1, 8)

733

views

Textbook Question

Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3

y = - (1/2)x + 2

734

views

Textbook Question

Solve each equation in Exercises 15–34 by the square root property. (x + 2)² = 25

949

views

Textbook Question

Solve and check each linear equation. 2(x - 1) + 3 = x - 3(x + 1)

1020

views

Textbook Question

You invested \$20,000 in two accounts paying 1.45% and 1.59% annual interest. If the total interest earned for the year was \$307.50, how much was invested at each rate?

827

views