Textbook Question
What is an identity equation? Give an example.
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 123
List all numbers that must be excluded from the domain of each rational expression. 3/(2x2 + 4x - 9)
Verified step by step guidance1
Identify the domain of a rational expression: The domain consists of all real numbers except those that make the denominator equal to zero. For this problem, the denominator is 2x^2 + 4x - 9.
Set the denominator equal to zero to find the values of x that must be excluded: 2x^2 + 4x - 9 = 0.
Simplify the quadratic equation: Divide through by 2 to make the equation simpler, resulting in x^2 + 2x - 4.5 = 0.
Solve the quadratic equation using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a = 1, b = 2, and c = -4.5.
Calculate the discriminant (b^2 - 4ac) and substitute the values into the quadratic formula to find the two solutions for x. These solutions are the values that must be excluded from the domain.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial because they can have restrictions on their domain, specifically where the denominator equals zero, as division by zero is undefined.
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Rationalizing Denominators
Finding the Domain
The domain of a rational expression consists of all the possible input values (x-values) that do not make the denominator zero. To find the domain, one must solve the equation set by the denominator equal to zero and exclude those values from the domain.
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Factoring Polynomials
Factoring polynomials is the process of breaking down a polynomial into simpler components (factors) that, when multiplied together, give the original polynomial. This is essential for identifying the values that make the denominator zero, as it allows for easier solving of the equation.
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Related Practice
Textbook Question
When the sum of 6 and twice a positive number is subtracted from the square of the number, 0 results. Find the number.
Textbook Question
What is a conditional equation? Give an example.
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Textbook Question
Find all values of x satisfying the given conditions. y1 = - x2 + 4x - 2, y2 = - 3x2 + x - 1, and y1 - y2 = 0
Textbook Question
Find all values of x satisfying the given conditions. y1 = 2x2 + 5x - 4, y2 = - x2 + 15x - 10, and y1 - y2 = 0
Textbook Question
What is an inconsistent equation? Give an example.