Solve the equation. (Enter your answers as a comma-separated list.)
5x^2e^(−2x) − 3x^3e^(−2x) = 0
well, geez, just factor out the e^(-2x):
e^(-2x) (5x^2-3x^3) = 0
e^(-2x) x^2 (5-3x) = 0
That help?
To solve the equation 5x^2e^(-2x) - 3x^3e^(-2x) = 0, we can factor out the common term e^(-2x) from both terms on the left-hand side:
e^(-2x)(5x^2 - 3x^3) = 0
Now, set each factor equal to zero:
e^(-2x) = 0 (equation 1)
and
5x^2 - 3x^3 = 0 (equation 2)
Let's solve equation 1 first. Since e^(anything) is always positive and never equals zero, there is no solution for equation 1.
Now let's solve equation 2:
5x^2 - 3x^3 = 0
Factor out an x^2 term:
x^2(5 - 3x) = 0
Now set each factor equal to zero:
x^2 = 0
This implies x = 0.
and
5 - 3x = 0
-3x = -5
x = 5/3
Therefore, the solutions to the equation 5x^2e^(-2x) - 3x^3e^(-2x) = 0 are x = 0 and x = 5/3.