Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 83

Chapter 2, Problem 83

Solve each equation in Exercises 83–108 by the method of your choice. 2x² - x = 1

Verified step by step guidance

Rewrite the equation in standard quadratic form by moving all terms to one side of the equation:

2x^2 - x - 1 = 0

Identify the coefficients of the quadratic equation in the form

ax^2 + bx + c = 0

. Here,

a = 2

b = -1

, and

c = -1

Choose a method to solve the quadratic equation. For example, you can use the quadratic formula:

x = \(\frac{-b \pm \sqrt{b^2 - 4ac}\)}{2a}

Substitute the values of

a

b

, and

c

into the quadratic formula:

x = \(\frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-1)}\)}{2(2)}

Simplify the expression under the square root (the discriminant) and then simplify the entire formula to find the two possible solutions for

x

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. These equations can be solved using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the standard form and properties of quadratic equations is essential for solving them effectively.

Factoring

Factoring involves rewriting an expression as a product of its factors. For quadratic equations, this often means expressing the equation in the form (px + q)(rx + s) = 0. This method is particularly useful when the quadratic can be easily factored, allowing for straightforward solutions by setting each factor equal to zero.

The Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a method to find the roots of any quadratic equation. It is derived from completing the square and is applicable even when the equation cannot be factored easily. Understanding how to apply this formula is crucial for solving quadratic equations, especially when dealing with complex or irrational solutions.

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Related Practice

Textbook Question

In Exercises 59–94, solve each absolute value inequality. - 2|x - 4| ≥ - 4

Textbook Question

Solve each absolute value inequality. - 4|1 - x| < - 16

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Textbook Question

Solve each equation in Exercises 83–108 by the method of your choice. $5 x^{2} + 2 = 11 x$

Textbook Question

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. $∣ x^{2} - 6 ∣ = ∣ 5 x ∣$

Textbook Question

Compute the discriminant. Then determine the number and type of solutions for the given equation. x² - 3x - 7 = 0

Textbook Question

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Solve each equation in Exercises 83–108 by the method of your choice. 2x2 - x = 1

Verified video answer for a similar problem:

Key Concepts

Quadratic Equations

Factoring

The Quadratic Formula

Solve each equation in Exercises 83–108 by the method of your choice. 2x² - x = 1