It isn't necessary to do a secure wipe of the HD which is done to make data irretrievable and has already been done twice now since the sale of computer. If going through the troubleshooting steps given here doesn't help then using the Diskpart "Clean" command is sufficient and less wearing on the HD.
This is a reinstall of the factory OEM Windows 7 Home Premium which originally came on the computer?
Where did seller obtain the installation DVD to do the reinstall? If he burned it himself, Confirm the ISO integrity and burn another using ImgBurn at 4x speed, or write the ISO to flash stick using this tool with Windows 7 in the dropdown menu:
Universal USB Installer – Easy as 1 2 3, then boot it under USB or HD's.
If you have booted the flash stick to do successful installs elsewhere then make sure it boots correctly on this machine: tap the eMachines F10 one-time BIOS Boot Menu key to boot it if it doesn't work to set it first to boot in BIOS setup Boot Order.
Unplug all other HD's and peripherals.
Once installer boots, follow these steps to get a perfect reinstall of factory OEM:
Reinstalling Windows 7
Check for newer BIOS update on eMachines website. If none, reset the CMOS:
Clear CMOS - 3 Ways to Clear the CMOS - Reset BIOS.
While in BIOS setup, set SATA controller to AHCI to try install. If this fails, try IDE mode.
Try formatting your own Primary partition marked Active with Diskpart using these Commands from DVD Repair console Command Prompt:
Select Disk 0 (use Windows 7 HD #) create partition primary format fs=ntfs label="Windows 7" Active exit
Test RAM using memtest86 for 5-6 passes or overnight:
RAM - Test with Memtest86+
Test your HD using maker's diagnostics extended CD scan:
HD Diagnostic
Report back progress for added steps if needed.