/*! 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 53 Use the One-to-One Property to s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 53

Use the One-to-One Property to solve the equation for \(x\). $$e^{x^{2}-3}=e^{2 x}$$

Short Answer

Expert verified
The solutions to the equation \(e^{x^{2}-3}=e^{2 x}\) are \(x=3\) and \(x=-1\).

Step by step solution

01

Apply One-to-One Property of Exponential Functions

Given that \( e^{x^{2}-3} = e^{2x} \). When the base of the exponentials is the same, we can equate the exponents. Therefore, we can write \(x^{2}-3 = 2x\).
02

Rearrange into Quadratic equation

Rearrange the equation into standard quadratic equation form ( \(ax^{2} + bx + c = 0\) ). Subtractiong \(2x\) from both sides gives: \(x^{2} - 2x - 3 = 0\) . Now, this is a quadratic equation which we can solve by factoring, completing the square or the quadratic formula.
03

Solve the quadratic equation

The quadratic equation \(x^{2} - 2x - 3 = 0\) can be factored into \((x-3)(x+1)=0\). The solutions of this equation are \(x=3\) or \(x=-1\), which are obtained when each of the factors is equal to zero.

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!

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 the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $g(x)=\log _{6} x$$

Function \(\quad\) Value $$ f(x)=\ln x \quad x=18.42$$

Writing Use your school's library, the Internet, or some other reference source to write a paper describing John Napier's work with logarithms.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$3 \ln 5 x=10$$

Solve the equation algebraically. Round your result to three decimal places. Verify your answer using a graphing utility. $$2 x \ln \left(\frac{1}{x}\right)-x=0$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision 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.