Textbook Question
Find all values of x satisfying the given conditions. y1 = (2x - 1)/(x2 + 2x - 8), y2 = 2/(x + 4), y3 = 1/(x - 2), and y1 + y2 = y3.
Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
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 67
In Exercises 65–70, perform the indicated operation(s) and write the result in standard form. (2 + i)2 - (3 - i)2
Verified step by step guidance1
Step 1: Recall the formula for squaring a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Apply this formula to \((2 + i)^2\), where \(a = 2\) and \(b = i\). Expand it as \(2^2 + 2(2)(i) + i^2\).
Step 2: Simplify \((2 + i)^2\) by calculating each term: \(2^2 = 4\), \(2(2)(i) = 4i\), and \(i^2 = -1\) (since \(i^2 = -1\) by definition of the imaginary unit). Combine these to get \(4 + 4i - 1\).
Step 3: Simplify \(4 + 4i - 1\) to \(3 + 4i\). This is the result of \((2 + i)^2\).
Step 4: Similarly, apply the formula for squaring a binomial to \((3 - i)^2\), where \(a = 3\) and \(b = -i\). Expand it as \(3^2 + 2(3)(-i) + (-i)^2\).
Step 5: Simplify \((3 - i)^2\) by calculating each term: \(3^2 = 9\), \(2(3)(-i) = -6i\), and \((-i)^2 = -1\). Combine these to get \(9 - 6i - 1\), which simplifies to \(8 - 6i\). Subtract \((3 - i)^2\) from \((2 + i)^2\) to get \((3 + 4i) - (8 - 6i)\), and simplify the real and imaginary parts separately.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas 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, 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.
Recommended video:
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, it is important to express the final result in this form to clearly identify the real and imaginary components. This helps in further calculations and interpretations in complex number theory.
Recommended video:
05:02Multiplying Complex Numbers
Binomial Expansion
Binomial expansion refers to the process of expanding expressions that are raised to a power, such as (a + b)^n. In the context of complex numbers, this involves applying the formula (x + y)^2 = x^2 + 2xy + y^2 to simplify expressions like (2 + i)^2 and (3 - i)^2. Mastery of this concept is crucial for accurately calculating the squares of binomials.
Recommended video:
Guided course
03:41Special Products - Cube Formulas
Related Practice
Textbook Question
In Exercises 61–66, find all values of x satisfying the given conditions. y1 = 5/(x + 4), y2 = 3/(x + 3), y3 = (12x + 19)/(x2 + 7x + 12). and y1 + y2 = y3.
Textbook Question
Solve each equation by completing the square.
5
views
Textbook Question
Solve each equation in Exercises 65–74 using the quadratic formula. x2 + 5x + 3 = 0
1
views
Textbook Question
Perform the indicated operation(s) and write the result in standard form. (8 + 9i)(2 - i) - (1 - i)(1 + i)
Textbook Question
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. 2|3x - 2| = 14
1
views