/*! 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 11 Find all complex-number solution... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 11

Find all complex-number solutions. $$ 5 y^{2}=30 $$

Short Answer

Expert verified
The solutions are \(y = \sqrt{6}\) and \(y = -\sqrt{6}\).

Step by step solution

01

Isolate the variable term

First, divide both sides of the equation by 5 to isolate the term with the variable. This will simplify the equation: \[5y^2 = 30\] Dividing by 5 gives: \[y^2 = 6\]
02

Take the square root of both sides

Next, take the square root of both sides of the equation to solve for y. Remember that taking the square root will yield both positive and negative solutions: \[y = \pm\sqrt{6}\]
03

Write the final solutions

The solutions to the equation are the positive and negative square roots of 6: \[y = \sqrt{6}\] and \[y = -\sqrt{6}\]

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.

complex numbers
Complex numbers can be a confusing concept, but we'll break it down simply. Complex numbers extend the real numbers by including where the square roots of negative numbers can have solutions. A complex number is in the form: \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\). In the provided exercise, the solutions to the equation \(5y^2 = 30\) are real, but understanding these solutions thoroughly includes knowing how they fit into the system of complex numbers.
square roots
Square roots are utilized in solving quadratic equations like \(5y^2 = 30\). To find the square root of a number \(n\), we look for a number \(x\) such that \(x^2 = n\). In our equation, after isolating the variable term, we end up with \(y^2 = 6\). Taking the square root of both sides gives us two solutions: \(y = \pm\sqrt{6}\). This indicates that both \(\sqrt{6}\) and \(-\sqrt{6}\) are solutions. The symbol \pm\ denotes 'plus or minus,' indicating both positive and negative roots.
isolating the variable
Isolating the variable is a fundamental step in solving equations. It means rearranging the equation to get the variable term alone on one side. In the provided exercise: \(5y^2 = 30\), we start by dividing both sides by 5, resulting in \(y^2 = 6\). This process simplifies our equation and sets us up for the next step of solving for \(y\). Whether solving equations involving complex numbers, square roots, or other functions, isolating the variable helps simplify and clarify the solution path.

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

Find an equation in slope鈥搃ntercept form of a line with the given characteristics. Slope: \(2 ;\) contains \((-3,7)\)

Solve. (Use \(\left.4.9 t^{2}+v_{0} t=s .\right)\) a) A life preserver is dropped from a helicopter at an altitude of \(75 \mathrm{m} .\) Approximately how long does it take the life preserver to reach the water? b) A coin is tossed downward with an initial velocity of \(30 \mathrm{m} / \mathrm{sec}\) from an altitude of \(75 \mathrm{m}\) Approximately how long does it take the coin to reach the ground? c) Approximately how far will an object fall in \(2 \sec ,\) if thrown downward at an initial velocity of \(20 \mathrm{m} / \mathrm{sec}\) from a helicopter?

Sump Pump. The lift distances for a Liberty 250 sump pump moving fluid at various flow rates are shown in the following table. \(\begin{array}{|c|c|}\hline \text { Gallons per } & {\text { Lift Distance }} \\\ {\text { Minute }} & {\text { (in feet) }} \\ \hline 10 & {21} \\ {20} & {18} \\ {40} & {8} \\ \hline\end{array}\) a) Let \(x\) represent the flow rate, in gallons per minute, and \(d(x)\) the lift distance, in feet. Find a quadratic function that fits the data. b) Use the function to find the lift distance for a flow rate of 50 gal per min.

Find the domain of each function. $$f(x)=\sqrt{x^{2}-4 x-45}$$

Solve each formula for the indicated letter. Assume that all variables represent positive numbers. \(a^{2}+b^{2}=c^{2},\) for \(b\) (Pythagorean formula in two dimensions)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Calculus

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.