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 + 7 = 2x + 7
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 75
Solve each equation by the method of your choice.
Verified step by step guidance1
Identify the given quadratic equation: \(3x^2 - 7x + 1 = 0\).
Recall that a quadratic equation in the form \(ax^2 + bx + c = 0\) can be solved using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Substitute the coefficients from the equation into the quadratic formula: \(a = 3\), \(b = -7\), and \(c = 1\), so the formula becomes \(x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}\).
Simplify inside the square root (the discriminant): calculate \(b^2 - 4ac = (-7)^2 - 4 \cdot 3 \cdot 1\).
Evaluate the expression under the square root and then compute the two possible values for \(x\) by applying the plus and minus signs in the quadratic formula.
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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 typically has two solutions, which can be real or complex numbers.
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Introduction to Quadratic Equations
Factoring and the Zero Product Property
Factoring involves rewriting a quadratic equation as a product of two binomials. The Zero Product Property states that if the product of two factors is zero, then at least one factor must be zero, allowing us to solve for the variable.
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Quadratic Formula
The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides a method to find the roots of any quadratic equation. It is especially useful when factoring is difficult or impossible, and the discriminant (b² - 4ac) determines the nature of the roots.
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Related Practice
Textbook Question
In Exercises 59–94, solve each absolute value inequality. |(2x + 2)/4| ≥ 2
Textbook Question
Exercises 73–75 will help you prepare for the material covered in the next section. Rationalize the denominator: (7 + 4√2)/(2 - 5√2).
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Textbook Question
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |2x - 1| + 3 = 3
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Textbook Question
In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation.
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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. 10x + 3 = 8x + 3
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