Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

The Quadratic Formula

College Algebra

1. Equations & Inequalities

The Quadratic Formula

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2:46 minutes

Problem 48

Textbook Question

See Exercise 47.(b)Which equation has two nonreal complex solutions?

Verified step by step guidance

insert step 1> Identify the type of equation you are dealing with. Typically, equations with complex solutions are quadratic equations of the form ax^2 + bx + c = 0.

insert step 2> Use the discriminant to determine the nature of the solutions. The discriminant is given by b^2 - 4ac.

insert step 3> Calculate the discriminant for the given equation. If the discriminant is less than zero, the equation has two nonreal complex solutions.

insert step 4> Verify that the discriminant is indeed negative, confirming the presence of complex solutions.

insert step 5> Conclude that the equation with a negative discriminant has two nonreal complex solutions.

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Key Concepts

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

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Nonreal complex solutions occur when the imaginary part is non-zero, indicating that the solutions cannot be represented on the real number line.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants and 'a' is not zero. The solutions to quadratic equations can be found using the quadratic formula, and the nature of the solutions (real or complex) is determined by the discriminant, given by b² - 4ac.

Discriminant

The discriminant of a quadratic equation, represented as D = b² - 4ac, is a key value that helps determine the nature of the roots. If D > 0, there are two distinct real solutions; if D = 0, there is one real solution (a repeated root); and if D < 0, the equation has two nonreal complex solutions, indicating that the graph of the equation does not intersect the x-axis.