Domanda del libro di testo
Find all values of x satisfying the given conditions. y1 = (2x - 1)/(x2 + 2x - 8), y2 = 2/(x + 4), y3 = 1/(x - 2), and y1 + y2 = y3.
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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 67
Solve each equation by completing the square.
Guida verificata passo dopo passo1
Start with the given quadratic equation: \(3x^2 - 12x + 11 = 0\).
Divide the entire equation by 3 to make the coefficient of \(x^2\) equal to 1: \(x^2 - 4x + \frac{11}{3} = 0\).
Isolate the constant term on one side: \(x^2 - 4x = -\frac{11}{3}\).
To complete the square, take half of the coefficient of \(x\) (which is \(-4\)), square it, and add it to both sides. Half of \(-4\) is \(-2\), and \((-2)^2 = 4\). So add 4 to both sides: \(x^2 - 4x + 4 = -\frac{11}{3} + 4\).
Rewrite the left side as a perfect square: \((x - 2)^2 = -\frac{11}{3} + \frac{12}{3}\), then simplify the right side.
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Completing the Square
Completing the square is a method 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. Understanding this form is essential for identifying coefficients and applying methods like completing the square correctly.
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04:34Converting Standard Form to Vertex Form
Isolating the Variable
Isolating the variable involves manipulating the equation to express the variable term alone on one side. In completing the square, this often means dividing through by the coefficient of x² and rearranging terms to prepare for forming a perfect square.
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05:28Equations with Two Variables
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