Textbook Question
Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. (3x+1)/3 - 13/2 = (1-x)/4
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 27
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 \( (x - a)^2 = k \), where \(a = 3\) and \(k = -5\). The square root property states that if \( (x - a)^2 = k \), then \( x - a = \pm \sqrt{k} \).
Apply the square root property to the equation: write \( x - 3 = \pm \sqrt{-5} \).
Since the square root of a negative number involves imaginary numbers, express \( \sqrt{-5} \) as \( \sqrt{5}i \), where \(i\) is the imaginary unit with the property \( i^2 = -1 \). So, rewrite the equation as \( x - 3 = \pm \sqrt{5}i \).
Isolate \(x\) by adding 3 to both sides: \( x = 3 \pm \sqrt{5}i \).
Write the final solution as two complex numbers: \( x = 3 + \sqrt{5}i \) and \( x = 3 - \sqrt{5}i \). These are the solutions to the equation using the square root property.
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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 not real but complex. The imaginary unit i is defined as √(-1), enabling the expression of roots of negative numbers as multiples of i, such as √(-5) = i√5.
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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 involves algebraic steps like adding or subtracting terms to simplify the equation to the form (x - a)^2 = b.
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Guided course
05:28Equations with Two Variables
Related Practice
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Solve each radical equation in Exercises 11–30. Check all proposed solutions. √(2x + 3) + √(x - 2) = 2