Domanda del libro di testo
In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 1 - x/2 > 4
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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 43
In Exercises 35–46, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial.
Guida verificata passo dopo passo1
Identify the coefficient of the linear term in the binomial. Here, the binomial is \(x^2 - \frac{2}{3}x\), so the coefficient of \(x\) is \(-\frac{2}{3}\).
Take half of the coefficient of \(x\). Half of \(-\frac{2}{3}\) is \(-\frac{1}{3}\).
Square the result from step 2. Squaring \(-\frac{1}{3}\) gives \(\left(-\frac{1}{3}\right)^2 = \frac{1}{9}\).
Add this squared value to the original binomial to form a perfect square trinomial: \(x^2 - \frac{2}{3}x + \frac{1}{9}\).
Write the trinomial as a squared binomial using the form \((x + a)^2 = x^2 + 2ax + a^2\). Here, it factors as \(\left(x - \frac{1}{3}\right)^2\).
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Perfect Square Trinomial
A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, typically in the form (a + b)^2 = a^2 + 2ab + b^2. Recognizing or creating such trinomials helps simplify factoring and solving quadratic expressions.
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Solving Quadratic Equations by Completing the Square
Completing the Square
Completing the square involves adding a constant term to a quadratic expression to form a perfect square trinomial. This constant is found by taking half the coefficient of the linear term, squaring it, and adding it to the expression, enabling easier factoring or solving.
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06:24Solving Quadratic Equations by Completing the Square
Factoring Quadratic Expressions
Factoring quadratic expressions means rewriting them as a product of binomials or squares of binomials. After completing the square, the trinomial can be factored into (x + d)^2 form, which simplifies solving equations or analyzing the function.
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06:08Solving Quadratic Equations by Factoring
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