/*! 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 41 \(k^{2}+8=-41\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 41

\(k^{2}+8=-41\)

Short Answer

Expert verified
k = \pm 7i.

Step by step solution

01

- Isolate the variable term

Subtract 8 from both sides of the equation to isolate the term with the variable: \[k^2 + 8 - 8 = -41 - 8\] Thus, the equation simplifies to \[k^2 = -49\].
02

- Take the square root of both sides

To solve for \(k\), take the square root of both sides of the equation: \[\sqrt{k^2} = \sqrt{-49}\]. Since the square root of a negative number introduces an imaginary unit \(i\), this becomes \[k = \pm 7i\].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Isolating the Variable
When solving quadratic equations, the first step usually involves isolating the variable term. This makes the equation simpler and easier to manage. In the exercise example, we start with the equation: \(k^2 + 8 = -41\). To isolate the term containing the variable \(k^2\), we subtract 8 from both sides of the equation. This step is important because it simplifies our equation into a standard form that's easier to solve: \(k^2 = -49\). By isolating \(k^2\), we remove any constants that could complicate further steps.Don't forget: always do the same operation to both sides of the equation to maintain equality.
Square Roots
After isolating the variable term in a quadratic equation, the next step usually involves taking the square root of both sides of the equation. In our example, after isolating \(k^2\), we obtained \(k^2 = -49\). Now, we need to take the square root of both sides: \(\sqrt{k^2} = \sqrt{-49}\). Remember, the square root of \(k^2\) gives us \(k\), which is what we want to solve for. But be cautious: when dealing with quadratic equations, taking the square root introduces two possible solutions - a positive and a negative one. So, \(\sqrt{k^2} = \pm \sqrt{-49}\) translates into \(k = \pm7i\)
Imaginary Numbers
Imaginary numbers come into play when dealing with the square roots of negative numbers. In our exercise, we encountered \sqrt{-49}. The concept of imaginary numbers helps us handle these kinds of situations. An imaginary number is represented by the symbol \(i\), where \(i = \sqrt{-1}\). Because \(i\) is the square root of -1, we can rewrite \(\sqrt{-49}\) as \(\sqrt{49} \times \sqrt{-1} = 7i\). Imaginary numbers extend our number system to solve equations that don't have solutions within the realm of real numbers. So, for \(k^2 = -49, k = \pm7i\).

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

\(\sqrt{3^{2}-4(2)(8)}\)

\(p^{2}+9 p+14=0\)

\(z^{2}-20 z+164=0\)

\((5-3 i)+(8-7 i)\)

\(v^{2}+2 v+14=0\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.