/*! 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 125 Solve the equation algebraically... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 125

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$2 x \ln x+x=0$$

Short Answer

Expert verified
The solution to the equation \(2x\ln x + x = 0\) is \(x = e^{-0.5}\), approximately equal to 0.607, after rounding to three decimal places.

Step by step solution

01

Rearrange Equation

First, rearrange the equation \(2x \ln x + x = 0\) to isolate terms with \(x\), which results in \(x(2\ln x + 1) = 0\).
02

Set each factor equal to Zero

Set each factor equal to zero, resulting in two separate equations. One is \(x=0\), and the other is \(2\ln x + 1 = 0\). The equation \(x=0\) is not a solution since \(\ln(0)\) is undefined. For the second equation, solve it by isolating \(\ln x\) then using exponentiation to solve for \(x\), resulting in \(x=e^{-0.5}\).
03

Verify the Solution Using a Graphing Utility

Plot the equation \(2x\ln x + x = 0\). As the x-value approaches the calculated solution, the y-value of the function should approach 0, thereby visually confirming the result.

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Most popular questions from this chapter

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{3} x+\log _{3}(x-8)=2$$

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