/*! 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 73 Solve the equation algebraically... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 73

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

Short Answer

Expert verified
The solutions to the equation \(2 x^{2} e^{2 x}+2 x e^{2 x}=0\) are \(x = 0\) and \(x = -1\).

Step by step solution

01

Factor out the Common term

Looking at the equation, it can be noticed that \(2x e^{2x}\) is a common term in both the terms on the left side of the equation. Hence, factor out \(2x e^{2x}\) from the equation. Therefore, \(2 x^{2} e^{2 x}+2 x e^{2 x}=0\) becomes \((2x e^{2x} )(x + 1) = 0\)
02

Separate the Equations

Once the equation is factored properly, for the product of two factors to be zero, either one or both must be zero. Therefore, set the two factors individually equal to zero to solve for \(x\). Thus, the two equations become \(2x e^{2x} = 0\) and \(x + 1 = 0\)
03

Solve the Equations

For the equation \(2x e^{2x} = 0\), \(x\) becomes 0 as any factor that multiplies the \(e^{2x}\) to be zero would be 0. For the equation \(x + 1 = 0\), solving for \(x\), we get \(x = -1\) by subtracting 1 from both sides.
04

Verify the Answers

To verify these answers, substitute these values of \(x\) back into the original equation and confirm if they satisfy the equation. Plugging \(x=0\) and \(x = -1\) into the original equation, the equation holds true, verifying these values are correct.

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