Textbook Question
In Exercises 37–52, perform the indicated operations and write the result in standard form. (3√-5)(- 4√-12)
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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 51
Solve each equation in Exercises 47–64 by completing the square.
Verified step by step guidance1
Start with the given quadratic equation: \(x^2 - 6x - 11 = 0\).
Move the constant term to the other side to isolate the \(x\) terms: \(x^2 - 6x = 11\).
To complete the square, take half of the coefficient of \(x\), which is \(-6\), divide by 2 to get \(-3\), then square it to get \((-3)^2 = 9\).
Add this square (9) to both sides of the equation to maintain equality: \(x^2 - 6x + 9 = 11 + 9\).
Rewrite the left side as a perfect square trinomial: \((x - 3)^2 = 20\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 adding and subtracting a specific value to both sides to create a binomial squared, making it easier to solve for the variable.
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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. Understanding this form helps identify coefficients needed for completing the square and applying the quadratic formula or other solving methods.
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04:34Converting Standard Form to Vertex Form
Solving Quadratic Equations
Solving quadratic equations means finding the values of the variable that satisfy the equation. Methods include factoring, completing the square, and using the quadratic formula, each useful depending on the equation's structure.
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Related Practice
Textbook Question
Write each English sentence as an equation in two variables. Then graph the equation. The y-value is two more than the square of the x-value.
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Textbook Question
Write each English sentence as an equation in two variables. Then graph the equation. y = 5 (Let x = -3, - 2, - 1, 0, 1, 2, and 3.)
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Textbook Question
Perform the indicated operations and write the result in standard form. (7-i)(2+3i)
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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. 3/(x + 4) - 7 = - 4/(x + 4)
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Textbook Question
In Exercises 51–58, solve each compound inequality. 6 < x + 3 < 8
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