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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 89
Chapter 2, Problem 89

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

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1
Rewrite the given equation \(x^2 - 2x = 1\) by moving all terms to one side to set the equation equal to zero: \(x^2 - 2x - 1 = 0\).
Identify the coefficients in the quadratic equation \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -2\), and \(c = -1\).
Apply the quadratic formula to solve for \(x\): \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-1)}}{2(1)}\).
Simplify the expression under the square root (the discriminant) and then simplify the entire expression to find the two possible values of \(x\).

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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 degree two, generally written as ax² + bx + c = 0. Solving such equations involves finding values of x that satisfy the equation, which can be done by factoring, completing the square, or using the quadratic formula.
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Rearranging Equations

Before solving, it is important to rewrite the equation in standard form (ax² + bx + c = 0). This involves moving all terms to one side of the equation and simplifying, which sets the stage for applying solution methods effectively.
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Methods of Solving Quadratic Equations

Common methods include factoring, completing the square, and the quadratic formula. Factoring works when the quadratic can be expressed as a product of binomials; completing the square transforms the equation into a perfect square trinomial; the quadratic formula provides a direct solution for any quadratic.
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Related Practice
Textbook Question

Solve each radical equation in Exercises 88–89. √ (2x-3) + x = 3

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Textbook Question

The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 4/(x - 2) + 3/(x + 5) = 7/(x + 5)(x - 2)

Textbook Question

In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).] y = 2(x + 2)^2 + 5(x + 2) - 3

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Textbook Question

The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 4x/(x + 3) - 12/(x - 3) = (4x2 + 36)/(x2 - 9)

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Textbook Question

In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).]

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y=x−2−x−1−6y = x^{-2} - x^{-1} - 6

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Textbook Question

In Exercises 59–94, solve each absolute value inequality. 1 < |2 - 3x|

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