Mac is not starting

Possible Fixes for El Capitan or Later

You should try each, one at a time, then test to see if the problem is fixed before going on to the next.

Be sure to backup your files before proceeding if possible.

1. Shutdown the computer, wait 30 seconds, restart the computer.
2. Resetting your Mac’s PRAM and NVRAM
3. Reset the System Management Controller (SMC)
4. Start the computer in Safe Mode, then restart normally. This is slower than a standard startup.
5. Repair the disk by booting from the Recovery HD. Immediately after the chime hold down the Command and R keys until the Utility Menu appears. Choose Disk Utility and click on the Continue button. Select the indented (usually, Macintosh HD) volume entry from the side list. Click on the First Aid button in the toolbar. Wait for the Done button to appear. Quit Disk Utility and return to the Utility Menu. Restart the computer from the Apple Menu.
6. Create a New User Account Open Users & Groups preferences. Click on the lock icon and enter your Admin password when prompted. On the left under Current User click on the Add [+] button under Login Options. Setup a new Admin user account. Upon completion log out of your current account then log into the new account. If your problems cease, then consider switching to the new account and transferring your files to it - Transferring files from one User Account to another.
7. Download and install theOS X El Capitan 10.11.6 ComboUpdate or 10.12.6 Combo Update or macOS High Sierra 10.13.2 Combo Update as needed.
8. Reinstall OS X by booting from the Recovery HD using the Command and R keys. When the Utility Menu appears select Reinstall OS X then click on the Continue button.
9. Erase and Install OS X Restart the computer. Immediately after the chime hold down the CommandandRkeys until the Apple logo appears. When the Utility Menu appears:

1. Select Disk Utility from the Utility Menu and click on Continue button.
2. When Disk Utility loads select the drive (out-dented entry) from the Device list.
3. Click on the Erase icon in Disk Utility's toolbar. A panel will drop down.
4. Set the Format type to Mac OS Extended (Journaled.)
5. Click on the Apply button, then wait for the Done button to activate and click on it.
6. Quit Disk Utility and return to the Utility Menu.
7. Select Reinstall OS X and click on the Continue button.

Try an SMC reset:

https://support.apple.com/en-us/HT201295

That's your call but you know the condition of your Mac and a used one is always a gamble. You might check with Apple's refurbished products which come with a warranty.

Refurbished Mac - Apple

Sorry to hear that. But if it's the motherboard there's not much else you can do.