Solve each equation using the quadratic formula. x2 - x - 1 = 0

Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 51

Chapter 2, Problem 51

Solve each equation using the quadratic formula. x² - x - 1 = 0

Verified step by step guidance

Identify the coefficients in the quadratic equation \(x^2 - x - 1 = 0\). Here, \(a = 1\), \(b = -1\), and \(c = -1\).

Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-1)}}{2(1)}\).

Simplify inside the square root: calculate the discriminant \(\Delta = b^2 - 4ac = (-1)^2 - 4(1)(-1)\).

Write the expression for \(x\) with the simplified discriminant under the square root, ready for further simplification and solving.

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Key Concepts

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

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the coefficients.

Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. It uses the coefficients a, b, and c to find the roots, including complex solutions when the discriminant is negative.

Discriminant

The discriminant, given by b² - 4ac, determines the nature of the roots of a quadratic equation. If positive, there are two distinct real roots; if zero, one real root; and if negative, two complex conjugate roots.

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