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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblem 41
Chapter 2, Problem 41

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. x2−7xx^2 - 7x

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1
Identify the coefficient of the linear term in the binomial. Here, the binomial is \(x^2 - 7x\), so the coefficient of \(x\) is \(-7\).
Take half of the coefficient of \(x\). Calculate \(\frac{-7}{2}\), which is \(-\frac{7}{2}\).
Square the result from step 2. Compute \(\left(-\frac{7}{2}\right)^2 = \frac{49}{4}\).
Add this squared value to the original binomial to form a perfect square trinomial: \(x^2 - 7x + \frac{49}{4}\).
Write the trinomial as a squared binomial using the form \(\left(x + a\right)^2 = x^2 + 2ax + a^2\). Here, it factors as \(\left(x - \frac{7}{2}\right)^2\).

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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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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 x-term, 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 a binomial squared. Once the perfect square trinomial is formed, it can be factored into (x + a)^2 or (x - a)^2, simplifying solving or further manipulation.
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Related Practice
Textbook Question

Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 3x/5 - (x - 3)/2 = (x + 2)/3

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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. 4/x = 5/2x + 3

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Textbook Question

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. x2 + 3x

Textbook Question

In Exercises 37–52, perform the indicated operations and write the result in standard form. (- 2 + √-4)2

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Textbook Question

In Exercises 35–54, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? E = mc2 for m

Textbook Question

Solve each equation with rational exponents in Exercises 31–40. Check all proposed solutions. (x2−x−4)34−2=6(x^2 - x - 4)^{\(\frac{3}{4}\)} - 2 = 6

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