Domanda del libro di testo
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
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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 75
Solve each equation by the method of your choice.
Guida verificata passo dopo passo1
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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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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6:29Factor Using Special Product Formulas
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.
Video consigliato:
06:36Solving Quadratic Equations Using The Quadratic Formula
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