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

Intro to Quadratic Equations

College Algebra

1. Equations & Inequalities

Intro to Quadratic Equations

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1:25 minutes

Problem 30

Textbook Question

Solve each equation using the square root property. See Example 2. x^2 = -400

Verified step by step guidance

Recognize that the equation is in the form \(x^2 = c\), where \(c = -400\).

To use the square root property, rewrite the equation as \(x^2 = -400\).

Take the square root of both sides of the equation, remembering to include both the positive and negative roots: \(x = \pm \sqrt{-400}\).

Since the square root of a negative number involves imaginary numbers, express \(\sqrt{-400}\) as \(\sqrt{400} \cdot \sqrt{-1}\).

Simplify \(\sqrt{400}\) to 20 and \(\sqrt{-1}\) to \(i\), giving the solution \(x = \pm 20i\).

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Key Concepts

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

Square Root Property

The square root property states that if x^2 = k, then x = ±√k. This property is essential for solving quadratic equations, as it allows us to isolate the variable by taking the square root of both sides. It is particularly useful when the equation is already in the form of a perfect square.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where i is the imaginary unit defined as √(-1). When solving equations like x^2 = -400, the square root of a negative number leads to complex solutions, which are crucial for understanding the complete set of solutions in algebra.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. They can have zero, one, or two real or complex solutions. Understanding the nature of these equations and their solutions is fundamental in algebra, especially when applying methods like the square root property.