Textbook Question
In Exercises 49–56, factor each perfect square trinomial. x^2−14x+49
Ch. P - Fundamental Concepts of Algebra
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 1, Problem 51
Rationalize the denominator.
Verified step by step guidance1
Identify the expression to rationalize: \(\frac{7}{\sqrt{5} - 2}\), where the denominator contains a radical.
To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{5} - 2\) is \(\sqrt{5} + 2\).
Set up the multiplication: \(\frac{7}{\sqrt{5} - 2} \times \frac{\sqrt{5} + 2}{\sqrt{5} + 2}\).
Multiply the numerators: \(7 \times (\sqrt{5} + 2)\), and multiply the denominators using the difference of squares formula: \((\sqrt{5})^2 - (2)^2\).
Simplify the denominator to \(5 - 4\), and write the expression as \(\frac{7(\sqrt{5} + 2)}{1}\), which completes the rationalization process.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rationalizing the Denominator
Rationalizing the denominator involves eliminating any irrational numbers, such as square roots, from the denominator of a fraction. This is done to simplify the expression and make it easier to work with or interpret.
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Conjugates of Binomials
The conjugate of a binomial expression a + b is a - b, and vice versa. Multiplying a binomial by its conjugate results in a difference of squares, which helps eliminate square roots in the denominator.
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Difference of Squares Formula
The difference of squares formula states that (a + b)(a - b) = a² - b². This property is used when multiplying by the conjugate to remove radicals from the denominator by turning it into a rational number.
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Related Practice
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