Textbook Question
In Exercises 15–26, use graphs to find each set. (- ∞, 5) ⋃ [1, 8)
Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 20
In Exercises 9–20, find each product and write the result in standard form. (5 - 2i)2
Verified step by step guidance1
Recognize that the expression (5 - 2i)^2 is a binomial squared. Use the formula for the square of a binomial: (a - b)^2 = a^2 - 2ab + b^2.
Identify the values of a and b in the binomial. Here, a = 5 and b = 2i.
Substitute the values of a and b into the formula: (5 - 2i)^2 = 5^2 - 2(5)(2i) + (2i)^2.
Simplify each term: 5^2 = 25, -2(5)(2i) = -20i, and (2i)^2 = 4i^2. Recall that i^2 = -1, so 4i^2 = 4(-1) = -4.
Combine the simplified terms: 25 - 20i - 4. Write the result in standard form as a + bi, where a is the real part and b is the imaginary part.
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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, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.
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Dividing Complex Numbers
Multiplication of Complex Numbers
To multiply complex numbers, you apply the distributive property (also known as the FOIL method for binomials) and combine like terms. When multiplying, remember that i^2 equals -1, which is crucial for simplifying the result. This process allows you to find the product of two complex numbers effectively.
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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 bi represents the imaginary part. Writing complex numbers in standard form is important for clarity and consistency in mathematical communication, especially when performing further operations.
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Related Practice
Textbook Question
Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = - (1/2)x + 2
Textbook Question
Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = -(1/2)x
Textbook Question
Find each product and write the result in standard form. (2 + 3i)2
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Textbook Question
Solve and check each linear equation. 2(x - 1) + 3 = x - 3(x + 1)
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?