Textbook Question
In Exercises 37–52, perform the indicated operations and write the result in standard form. √-64 - √-25
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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 37a
Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. x2 - 10x
Verified step by step guidance1
Identify the coefficient of the linear term (the term with x). In this case, the coefficient of x is -10.
Divide the coefficient of x by 2. This gives \(-10 \div 2 = -5\).
Square the result from the previous step. \((-5)^2 = 25\). This is the constant that should be added to the binomial to make it a perfect square trinomial.
Add the constant (25) to the binomial \(x^2 - 10x\), resulting in \(x^2 - 10x + 25\).
Factor the trinomial \(x^2 - 10x + 25\) as \((x - 5)^2\), which is the square of a binomial.
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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 expressed as the square of a binomial. It takes the form (a ± b)² = a² ± 2ab + b². Recognizing this structure is essential for transforming a given binomial into a perfect square trinomial by adding the appropriate constant.
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Completing the Square
Completing the square is a method used to convert a quadratic expression into a perfect square trinomial. This involves taking half of the coefficient of the linear term, squaring it, and adding it to the expression. This technique is crucial for determining the constant needed to complete the square in the given binomial.
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Factoring Quadratics
Factoring quadratics involves rewriting a quadratic expression as a product of its linear factors. For a perfect square trinomial, this means expressing it in the form (a ± b)². Understanding how to factor these expressions is important for simplifying and solving quadratic equations effectively.
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Related Practice
Textbook Question
In Exercises 35–54, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? A = (1/2)bh for b
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Textbook Question
Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 5 + (x - 2)/3 = (x + 3)/8
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Textbook Question
In Exercises 37–52, perform the indicated operations and write the result in standard form. 5√-16 + 3√-81
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Textbook Question
Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions. (x - 4)2/3 = 16
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Textbook Question
Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? V = (1/3)Bh for B
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