/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 76 Solve each equation. (All soluti... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 76

Solve each equation. (All solutions are nonreal complex numbers.) $$ x^{2}=-21 $$

Short Answer

Expert verified
x = \pm \sqrt{21} \cdot i\

Step by step solution

01

Understand the equation

The equation given is \(x^{2}=-21\). This implies that when \(x\) is squared, it equals \(-21\). Here, we need to find complex solutions since the square of a real number cannot be negative.
02

Set up the equation for complex solutions

Recall that \(-21\) can be represented as \(-21 = -1 \cdot 21\). Since \-1\ can be written as \i^{2}\ (where \i\ is the imaginary unit), we rewrite the equation as follows: \(x^{2} = 21 \cdot i^{2}\). Now, set \(x = \pm\ sqrt{21} \cdot i\).
03

Solve for \(x\)

Solving for \(x\), we take the square root of both sides: \(x = \pm\ sqrt{-21} = \pm \sqrt{21} \cdot i\). Hence, the solutions are \(x = \sqrt{21} \cdot i\) and \(x = -\sqrt{21} \cdot i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving equations
Solving equations is a fundamental aspect of algebra. When faced with an equation, the goal is to determine the value or values of the variable that make the equation true. In the given exercise, we are solving for the variable x in the equation \( x^{2} = -21 \). For regular equations involving real numbers, if you end up with a negative result when squaring, it indicates the need for complex numbers. This leads us to explore the realm of imaginary numbers and complex solutions.
imaginary unit
The imaginary unit, denoted as \( i \), is a crucial element when dealing with complex numbers. By definition, \( i \) is the square root of -1, which means \( i^{2} = -1 \). In the context of the exercise, recognizing that \( -21 \) can be broken down as \( 21 \times -1 \) helps us rewrite the equation to use the imaginary unit. This gives us \( x^{2} = 21 \times i^{2} \). From this point, we can solve for x by taking the square root of both sides, yet keeping in mind the properties of \( i \).
square root of negative numbers
Taking the square root of negative numbers is essential for finding complex solutions. With real numbers, the square root of a negative number is undefined. However, by introducing the imaginary unit \( i \), we can handle these cases. In the exercise, we look at \( x^{2} = -21 \) and recognize that this can be rewritten as \( x = \pm \, \sqrt{-21} \). Utilizing the imaginary unit, this becomes \( x = \pm \, \sqrt{21} \, i \). Hence, the solutions are \( x = \sqrt{21} \, i \) and \( x = -\sqrt{21} \, i \).
algebra
Algebra provides the tools and rules for manipulating equations to find solutions. Complex numbers are an extension of algebra that allows for solving equations involving the square roots of negative numbers. In this exercise, we use algebraic techniques to understand and solve \( x^{2} = -21 \). We utilize the property \( -1 = i^{2} \) to express the equation in terms of \( i \). Solving for \( x \) involves recognizing the role of the imaginary unit and calculating accordingly. Through algebra, we determine the solutions \( x = \sqrt{21} \, i \) and \( x = -\sqrt{21} \, i \), demonstrating the power and breadth of algebraic methods.

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