Textbook Question
In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 5x + 9 = 9(x + 1) - 4x
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 72
Without solving the given quadratic equation, determine the number and type of solutions.
Verified step by step guidance1
Rewrite the given quadratic equation in standard form \(ax^2 + bx + c = 0\). Start by moving all terms to one side: \(9x^2 + 3x - 2 = 0\).
Identify the coefficients: \(a = 9\), \(b = 3\), and \(c = -2\).
Calculate the discriminant using the formula \(\Delta = b^2 - 4ac\).
Substitute the values into the discriminant formula: \(\Delta = (3)^2 - 4(9)(-2)\).
Analyze the discriminant value: if \(\Delta > 0\), there are two distinct real solutions; if \(\Delta = 0\), there is one real repeated solution; if \(\Delta < 0\), there are two complex solutions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of a Quadratic Equation
A quadratic equation is typically written in the form ax² + bx + c = 0, where a, b, and c are constants. Converting the given equation into this form is essential for analyzing its properties, such as the number and type of solutions.
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Converting Standard Form to Vertex Form
Discriminant of a Quadratic Equation
The discriminant, given by Δ = b² - 4ac, determines the nature of the roots of a quadratic equation. If Δ > 0, there are two distinct real solutions; if Δ = 0, there is one real repeated solution; and if Δ < 0, there are two complex conjugate solutions.
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Types of Solutions for Quadratic Equations
Quadratic equations can have real or complex solutions. Real solutions occur when the graph of the quadratic intersects the x-axis, while complex solutions occur when it does not. Understanding this helps in interpreting the discriminant without solving the equation.
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Related Practice
Textbook Question
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |x + 1| + 5 = 3
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Textbook Question
In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 4x + 7 = 7(x + 1) - 3x
Textbook Question
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. 2|4 - (5/2)x| + 6 = 18
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Textbook Question
In Exercises 59–94, solve each absolute value inequality. |x - 1| ≥ 2
Textbook Question
Exercises 73–75 will help you prepare for the material covered in the next section. Multiply: (7 - 3x)(- 2 - 5x)