Textbook Question
Solve each equation in Exercises 96–102 by the method of your choice. x^3 + 2x^2 = 9x + 18
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 99
Solve each equation in Exercises 83–108 by the method of your choice.
Verified step by step guidance1
Identify the type of equation: This is a quadratic equation in the form \(x^2 - 6x + 13 = 0\), where \(a = 1\), \(b = -6\), and \(c = 13\).
Calculate the discriminant using the formula \(\Delta = b^2 - 4ac\). Substitute the values to get \(\Delta = (-6)^2 - 4(1)(13)\).
Evaluate the discriminant to determine the nature of the roots: if \(\Delta > 0\), there are two real roots; if \(\Delta = 0\), one real root; if \(\Delta < 0\), two complex roots.
Since the discriminant is less than zero, use the quadratic formula to find the complex roots: \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\).
Substitute \(a\), \(b\), and \(\Delta\) into the quadratic formula and simplify to express the solutions in terms of real and imaginary parts.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Equations
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the discriminant.
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Introduction to Quadratic Equations
Discriminant and Nature of Roots
The discriminant, given by b² - 4ac, determines the nature of the roots of a quadratic equation. If it is positive, there are two distinct real roots; if zero, one real root; and if negative, two complex conjugate roots.
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Solving Quadratic Equations Using the Quadratic Formula
The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation. It is especially useful when factoring is difficult or impossible, and it accounts for all types of roots based on the discriminant.
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Related Practice
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Textbook Question
In Exercises 99–106, solve each equation. [(3 + 6)2 ÷ 3] × 4 = - 54 x
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