Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7. x² - 6x = -7
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Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem R.6.67
Solve each quadratic equation using the quadratic formula. See Example 7.
x² - x - 1 = 0
Verified step by step guidance1
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: \(x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \times 1 \times (-1)}}{2 \times 1}\).
Simplify inside the square root (the discriminant): calculate \(b^2 - 4ac = (-1)^2 - 4 \times 1 \times (-1)\).
Evaluate the entire expression under the square root and then write the two possible solutions for \(x\) using the \(\pm\) sign.
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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, b, and c are constants and a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the discriminant.
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Introduction to Quadratic Equations
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.
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Discriminant
The discriminant, given by Δ = b² - 4ac, determines the nature of the roots of a quadratic equation. If Δ > 0, there are two distinct real roots; if Δ = 0, one real root; and if Δ < 0, two complex conjugate roots.
Related Practice
Textbook Question
Solve each inequality. Give the solution set using interval notation. See Example 10. -9 ≤ x + 5 ≤ 15
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Textbook Question
Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -2x - 2 ≤ 1 + x
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Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7. x² - 2x - 2 = 0
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Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5. 5x² - 3x - 2 = 0
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Textbook Question
Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 4x + 7 ———— ≤ 2x + 5 -3
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