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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 55
Chapter 2, Problem 55

Solve each equation in Exercises 47–64 by completing the square. x2−5x+6=0x^2 - 5x + 6 = 0

Verified step by step guidance
1
Start with the given quadratic equation: \(x^2 - 5x + 6 = 0\).
Move the constant term to the other side to isolate the quadratic and linear terms: \(x^2 - 5x = -6\).
To complete the square, take half of the coefficient of \(x\), which is \(-5\), divide by 2 to get \(-\frac{5}{2}\), then square it to get \(\left(-\frac{5}{2}\right)^2 = \frac{25}{4}\).
Add \(\frac{25}{4}\) to both sides of the equation to maintain equality: \(x^2 - 5x + \frac{25}{4} = -6 + \frac{25}{4}\).
Rewrite the left side as a perfect square trinomial: \(\left(x - \frac{5}{2}\right)^2 = -6 + \frac{25}{4}\). From here, you can simplify the right side and solve for \(x\) by taking the square root of both sides.

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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 create a binomial squared, making it easier to solve for the variable.
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Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0. Understanding its structure is essential for applying methods like factoring, completing the square, or using the quadratic formula to find the roots.
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Solving Equations Using Square Roots

Once a quadratic equation is written as a perfect square equal to a constant, you can solve it by taking the square root of both sides. This step introduces both positive and negative roots, which are critical for finding all solutions.
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Related Practice
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/(2x - 2) + 1/2 = 2/(x - 1)

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 + 2) + 2/(x - 2) = 8/(x + 2)(x - 2)

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Textbook Question

Solve each equation in Exercises 47–64 by completing the square. x2+4x+1=0x^2 + 4x + 1 = 0

Textbook Question

Solve each equation in Exercises 41–60 by making an appropriate substitution. (x - 5)2 - 4(x - 5) - 21 = 0

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Textbook Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? 1/R = 1/R1 + 1/R2 for R1

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Textbook Question

In Exercises 51–58, solve each compound inequality. - 11 < 2x - 1 ≤ - 5