Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility.
e−^8x − 6xe^−8x = 0
x =
I need help solving this problem.
To solve the equation algebraically, we can start by factoring out e^(-8x) from both terms:
e^(-8x)(1 - 6x) = 0
To find the solution, we set each factor equal to 0:
e^(-8x) = 0 and 1 - 6x = 0
The first equation, e^(-8x) = 0, has no solutions because e^(-8x) is never equal to 0.
The second equation, 1 - 6x = 0, can be solved as follows:
1 - 6x = 0
6x = 1
x = 1/6
Therefore, the solution to the equation e^(-8x) - 6xe^(-8x) = 0 is x = 1/6.
To verify this answer using a graphing utility:
1. Plot the given equation y = e^(-8x) - 6xe^(-8x).
2. Use the graphing utility to find the x-coordinate where the graph intersects the x-axis.
3. Round the x-coordinate to three decimal places.
4. Confirm that the rounded x-coordinate matches our algebraic solution of x = 1/6.
By following these steps, we can confirm that the solution to the equation e^(-8x) - 6xe^(-8x) = 0 is x = 0.167 (rounded to three decimal places).