/*! 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 76 Solve the equation algebraically... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 76

Solve the equation algebraically. Round your 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 \(e^{-2 x}-2 x e^{-2 x}=0\) is \(x = 0.5\).

Step by step solution

01

Simplify the Equation

Notice that both terms in the equation \(e^{-2 x}-2 x e^{-2 x}=0\) contain a common factor of \(e^{-2x}\). Factoring out the common element will simplify the equation. After factoring, we obtain: \(e^{-2x}(1-2x) = 0\).
02

Set Each Factor to Zero

In order for a product to be zero, either factor must be zero. This gives two scenarios to solve: \(e^{-2x} = 0\) and \(1-2x = 0\). The first equation has no solution because the exponential function always has positive values. Solving the second equation \(1-2x = 0\) for \(x\), we obtain \(x = 0.5\). This is our solution.
03

Verify the Solution Graphically

Plot the given equation \(e^{-2 x}-2 x e^{-2 x}\) on a graph. The solution to the equation is the x-value(s) at which the graph crosses the x-axis (y=0). When you plot the function, you will observe that it crosses the x-axis at \(x = 0.5\), confirming that the calculation in Step 2 is correct.

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