Textbook Question
In Exercises 21–28, divide and express the result in standard form. 5i/(2 - i)
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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 25
Solve each equation in Exercises 15–34 by the square root property.
Verified step by step guidance1
Recognize that the equation is in the form \( (x + 3)^2 = -16 \), which is suitable for applying the square root property. The square root property states that if \( A^2 = B \), then \( A = \pm \sqrt{B} \).
Apply the square root property to both sides of the equation: \( x + 3 = \pm \sqrt{-16} \).
Recall that the square root of a negative number involves imaginary numbers. Express \( \sqrt{-16} \) as \( \sqrt{16} \times \sqrt{-1} \), which simplifies to \( 4i \), where \( i \) is the imaginary unit.
Rewrite the equation using this simplification: \( x + 3 = \pm 4i \).
Isolate \( x \) by subtracting 3 from both sides: \( x = -3 \pm 4i \). This gives the two complex solutions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Square Root Property
The square root property states that if an equation is in the form (x + a)^2 = b, then x + a = ±√b. This allows solving quadratic equations by isolating the squared term and taking the square root of both sides, considering both positive and negative roots.
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Imaginary Roots with the Square Root Property
Complex Numbers and Imaginary Unit
When the equation involves the square root of a negative number, solutions are complex numbers. The imaginary unit i is defined as √(-1), enabling the expression of roots of negative numbers as multiples of i, such as √(-16) = 4i.
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03:31Introduction to Complex Numbers
Isolating the Variable
Before applying the square root property, the equation must be manipulated to isolate the squared term on one side. This often involves adding or subtracting constants and ensures the equation is in the correct form for taking square roots.
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Guided course
05:28Equations with Two Variables
Related Practice
Textbook Question
Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = |x| + 1
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Textbook Question
Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 2x/3 = 6 - x/4
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Textbook Question
In Exercises 15–26, use graphs to find each set. [3, ∞) ⋃ (6, ∞)
Textbook Question
In Exercises 1–26, solve and check each linear equation. 16 = 3(x - 1) - (x - 7)
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Textbook Question
A rectangular soccer field is twice as long as it is wide. If the perimeter of the soccer field is 300 yards, what are its dimensions?