Textbook Question
Solve each equation in Exercises 41–60 by making an appropriate substitution. x - 13√x + 40 = 0
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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 45
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.
Verified step by step guidance1
Identify the coefficient of the linear term in the binomial. Here, the binomial is \(x^2 - \frac{1}{3}x\), so the coefficient of \(x\) is \(-\frac{1}{3}\).
To complete the square, take half of the coefficient of \(x\). Calculate \(\frac{1}{2} \times \left(-\frac{1}{3}\right) = -\frac{1}{6}\).
Square the result from the previous step to find the constant to add: \(\left(-\frac{1}{6}\right)^2 = \frac{1}{36}\).
Add this constant to the original binomial to form the perfect square trinomial: \(x^2 - \frac{1}{3}x + \frac{1}{36}\).
Write the trinomial as a squared binomial using the value found in step 2: \(\left(x - \frac{1}{6}\right)^2\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 this form helps in rewriting and factoring quadratics efficiently.
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Solving Quadratic Equations by Completing the Square
Completing the Square
Completing the square involves adding a constant 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.
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Factoring Quadratic Expressions
Factoring quadratic expressions means rewriting them as a product of binomials. Once a perfect square trinomial is formed, it can be factored as the square of a binomial, simplifying solving or further manipulation.
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Related Practice
Textbook Question
Solve and check: 24 + 3 (x + 2) = 5(x − 12).
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Textbook Question
Use the graph to a. determine the x-intercepts, if any; b. determine the y-intercepts, if any. For each graph, tick marks along the axes represent one unit each.
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Textbook Question
Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 7/2x - 5/3x = 22/3
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Textbook Question
In Exercises 45–47, solve each formula for the specified variable. vt + gt^2 = s for g
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Textbook Question
In Exercises 37–52, perform the indicated operations and write the result in standard form. (- 8 + √-32)/24
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