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Chapter 3: Problem 124

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

Short Answer

Expert verified
The solution to the equation is x = 0.5.

Step by step solution

01

Simplify the Equation

Start the problem by trying to simplify it as much as possible, while observing the equation in question \(e^{-2x} - 2x e^{-2x} = 0\), it can be noticed that there is a common factor which can be factored out to simplify the equation. The common factor here is \(e^{-2x}\). So, the equation simplifies to \(e^{-2x} (1 - 2x) = 0\).
02

Equating factors to zero

This is the step where factoring comes in handy. Since a product of factors equals zero, then at least one of the factors must be equal to zero. Set each factor equal to zero: \(e^{-2x} = 0\) and \(1 - 2x = 0\).
03

Solve the First Equation

On solving the equation \(e^{-2x} = 0\), it is impossible because the exponential function is always positive, hence there are no real solutions to this equation.
04

Solve the Second Equation

On the other hand, by solving the other equation \(1 - 2x = 0\) for x, we have that x equals \(0.5\).
05

Checking the solution

Finally, substitute \(0.5\) into the original equation to check if it is valid . If the left hand side equals the right hand side of the equation, then the solution is correct.

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Most popular questions from this chapter

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