Vint is testing ceiling fans in a factory. For one of the tests, he switches the fan on, and after it attains a maximum speed of 500 rotations per minute (rpm), he switches the fan off and records the time it takes for the fan to stop spinning. The given equation models Vint's test, where [tex]$x$[/tex] represents time in seconds and [tex]$y$[/tex] represents the speed in rotations per minute.

[tex] y = -5x^2 + 100x [/tex]

The equation has been graphed as shown.

To solve the problem of finding when the fan completely stops, we need to determine the times when the speed of the fan, modeled by the equation [tex]$ y = -5x^2 + 100x $[/tex], is zero. This is because the fan stops completely when its speed is zero.

Here are the steps to find the solution:

1. Identify the Equation: We start with the equation representing the speed of the fan:
[tex]\[
y = -5x^2 + 100x
\][/tex]

2. Set the Speed to Zero: Since we want to find when the fan stops, we set [tex]$ y $[/tex] equal to zero:
[tex]\[
0 = -5x^2 + 100x
\][/tex]

3. Factor the Equation: We factor out the common term, which is [tex]$ 5x $[/tex], from the equation:
[tex]\[
0 = 5x(-x + 20)
\][/tex]

4. Solve for [tex]$ x $[/tex]: Set each factor equal to zero to solve for [tex]$ x $[/tex]:
- For the first factor [tex]$ 5x = 0 $[/tex], solving gives:
[tex]\[
x = 0
\][/tex]
- For the second factor [tex]$-x + 20 = 0$[/tex], solving gives:
[tex]\[
x = 20
\][/tex]

5. Interpret the Solutions: The solutions [tex]$ x = 0 $[/tex] and [tex]$ x = 20 $[/tex] represent the times, in seconds, at which the fan's speed is zero.

- At [tex]$ x = 0 $[/tex], this is the moment when the fan is just starting (and hasn't sped up yet).
- At [tex]$ x = 20 $[/tex], this is when the fan has returned to zero speed and has completely stopped.

Therefore, the fan completely stops after 20 seconds.