Domanda del libro di testo
In Exercises 59–94, solve each absolute value inequality. |3x - 8| > 7
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 2, Problema 72
Without solving the given quadratic equation, determine the number and type of solutions.
Guida verificata passo dopo passo1
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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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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04:34
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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04:11The Discriminant
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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05:35Introduction to Quadratic Equations
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