The laptop screen doesn't turn on

The first thing to do is to verify that the laptop is booting up and not just running the fan.

Observing the drive activity LED would be a good indicator. It will be very much on as soon as you turn it on, then gradually slow to almost stop blinking.

Assuming the laptop is booting up, the first thing to check would be to do a flashlight test to see if it is a backlight or video issue.

In my experience, a common problem with HP laptops is a loose display cable.

That requires opening up the unit. On some HPs, you can gain access to the video connector by removing the keyboard.

If there is a port to connect an external display/TV, try that first. If that works, it doesn't eliminate the video module.

If it is neither a backlight or a loose connector, then it is either the video module or the display.

The video module may be integral to the motherboard. That would be a MAJOR repair and likely not economical.