Chapter 3.4, Problem 76E

To determine

To find : solution to the equation algebraically and verify the result graphically.

Answer to Problem 76E

The solution to the equation $e^{-2x}-2x{e}^{-2x}=0$ are $\underset{\_}{\boxed{x=1/2}}$

Explanation of Solution

Given information: $e^{-2x}-2x{e}^{-2x}=0$

Concept Involved:

The word “solve” means the process of finding the value of x that makes the equation true.

In order to solve we need to undo whatever is done to x .

The solution to the equation $f\left(x\right)=g\left(x\right)$ graphically is the x value of the point of intersection of where the two graphs meet.

Formula Used:

(i)Zero factor property: If $a\cdot b=0$ , then $a=0\left(or\right)b=0$

Calculation:

Factor the Greatest Common Factor in the left side of the equation.

  $e^{-2x}\left(1-2x\right)=0$

Use the zero factor property to set each factor to zero

  $e^{-2x}=0$▶ 1^stequation

  $1-2x=0$▶ 2^ndequation

Solving the equation $e^{-2x}=0$

• There is no value of x that makes the above equation TRUE, so No solution

Solving the equation $1-2x=0$

• By subtracting1 on both sides, then
• By Simplifying on both sides, then
• By dividing -2 on both sides, and then
• By simplifying fraction on both sides of the equation
$1-2x-1=0-1$

  $-2x=-1$

  $-2x/-2=-1/-2$

  $x=1/2$

Check for extraneous solution by substituting the result in the original equation.

  $x=1/2$ makes the original equation TRUE.

Graph:

  

Interpretation:

Setting up left side of the equation $f\left(x\right)=e^{-2x}-2x{e}^{-x}$ and right side of the equation $g\left(x\right)=0$ .

The xcoordinate of point of intersection of two graphs is the solution to the equation $e^{-2x}-2x{e}^{-2x}=0$

Conclusion:

The solution to the equation $e^{-2x}-2x{e}^{-2x}=0$ are $\underset{\_}{\boxed{x=1/2}}$