Textbook Question
Write each complex number in standard form. (2 + 3i)3
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 64
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 binomial given: \(x^2 + 20x\).
Recall that to complete the square, we add a constant term to make the expression a perfect square trinomial. The formula for the constant to add is \(\left(\frac{b}{2}\right)^2\), where \(b\) is the coefficient of \(x\).
Calculate \(\left(\frac{20}{2}\right)^2 = 10^2 = 100\). This is the constant that should be added to the binomial.
Write the perfect square trinomial by adding 100: \(x^2 + 20x + 100\).
Factor the trinomial as a square of a binomial: \(\left(x + 10\right)^2\).
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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 (x + a)^2 = x^2 + 2ax + a^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 term to a quadratic expression to form a perfect square trinomial. This constant is found by taking half the coefficient of x, then squaring it, which allows the expression to be factored as a binomial squared.
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Factoring Quadratic Expressions
Factoring quadratic expressions means rewriting them as a product of binomials or squares of binomials. Once a perfect square trinomial is formed, it can be factored into (x + a)^2, simplifying solving or further manipulation.
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Related Practice
Textbook Question
In Exercises 59–94, solve each absolute value inequality. |2(x - 1) + 4| ≤ 8
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Textbook Question
Solve each equation in Exercises 47–64 by completing the square. 3x2 - 5x - 10 = 0
Textbook Question
In Exercises 59–94, solve each absolute value inequality. |3x + 5| < 17
Textbook Question
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |2x - 1| = 5
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Textbook Question
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |x - 2| = 7
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