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Ch. 1 - Equations and Inequalities
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 1 - Equations and InequalitiesProblem 2
Chapter 2, Problem 2

Match the equation in Column I with its solution(s) in Column II. x2 = -25
Matching exercise with equations involving squares and their corresponding solutions including real and imaginary numbers.

Verified step by step guidance
1
Recognize that the equation given is \(x^2 = -25\), which is a quadratic equation where the square of \(x\) equals a negative number.
Recall that in the set of real numbers, the square of any real number is always non-negative, so \(x^2 = -25\) has no real solutions.
To find solutions, consider the complex number system where \(i\) is defined as the imaginary unit with the property \(i^2 = -1\).
Rewrite the equation as \(x^2 = -25 = 25 \times (-1) = 25i^2\), then take the square root of both sides to get \(x = \pm \sqrt{25i^2}\).
Simplify the square root to \(x = \pm 5i\), which are the two complex solutions to the equation.

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Key Concepts

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

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. Solving them involves finding values of x that satisfy the equation, often using factoring, completing the square, or the quadratic formula.
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Solving Quadratic Equations by Factoring

Complex Numbers and Imaginary Unit

When a quadratic equation has no real solutions, such as x² = -25, solutions involve complex numbers. The imaginary unit i is defined as √-1, allowing expressions like √-25 to be written as 5i, representing imaginary solutions.
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Introduction to Complex Numbers

Square Roots of Negative Numbers

Taking the square root of a negative number is not possible within the real numbers. Instead, it introduces imaginary numbers, where √-a = i√a for a > 0. This concept is essential for understanding solutions to equations like x² = -25.
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Square Roots of Negative Numbers
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