Domanda del libro di testo
In Exercises 101–102, find an equation for f^(-1)(x). Then graph f and f^(-1) in the same rectangular coordinate system. f(x) = 1 - x^2, x ≥ 0.
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Ch. 2 - Functions and Graphs
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 3, Problema 100
Solve by completing the square: 2x² – 5x + 1 = 0.
Guida verificata passo dopo passo1
Start by dividing the entire equation by the coefficient of \(x^2\), which is 2, to make the coefficient of \(x^2\) equal to 1. This gives: \(x^2 - \frac{5}{2}x + \frac{1}{2} = 0\).
Next, move the constant term to the right side of the equation: \(x^2 - \frac{5}{2}x = -\frac{1}{2}\).
To complete the square, take half of the coefficient of \(x\), which is \(-\frac{5}{2}\), divide it by 2 to get \(-\frac{5}{4}\), then square it to get \(\left(-\frac{5}{4}\right)^2 = \frac{25}{16}\). Add this value to both sides of the equation.
Rewrite the left side as a perfect square trinomial: \(\left(x - \frac{5}{4}\right)^2 = -\frac{1}{2} + \frac{25}{16}\).
Simplify the right side by finding a common denominator and combining the terms, then solve for \(x\) by taking the square root of both sides and isolating \(x\).
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Completing the Square Method
Completing the square is a technique used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves creating a binomial squared expression on one side, making it easier to solve for the variable by taking square roots.
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Solving Quadratic Equations by Completing the Square
Quadratic Equation Standard Form
A quadratic equation is typically written in the form ax² + bx + c = 0, where a, b, and c are constants. Recognizing this form is essential before applying methods like completing the square, as it guides the steps needed to isolate and manipulate terms.
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04:34Converting Standard Form to Vertex Form
Isolating the Variable
Isolating the variable involves rearranging the equation so that the variable term is alone on one side, often requiring division or factoring out coefficients. This step is crucial in completing the square to simplify the equation and prepare it for taking square roots.
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05:28Equations with Two Variables
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