Re: Problem "No root file system is defined" during installation process
NO. There is no problem in true dual-boot. WUBI was meant for Windows Users to 'TRY Ubuntu' before doing a 'true dual-boot'.
Wubi install inside windows is just like any software application you install and run in Windows. You can remove it from Windows using Add/Remove software or something similar.
Since you tried installing then there is possiblity that you may have to remove it first from Windows, Here's a how to: https://wiki.ubuntu.com/WubiGuide#Uninstallation
Does your HP have Windows8 or Windows7? Pre-installed? If you have pre-installed Win8 then you could be booting in UEFI mode. This needs a few extra steps compared to 'Legacy BIOS boot' mode.
A screen shot of your Hard Disk and its partitions taken from Windows Disk Management tool will help us help you.
Download Ubuntu.iso and burn it to a USB/DVD. Boot with it and "Try Ubuntu without Installling", this will load Ubuntu, wait till the desktop is fully loaded. Open Terminal (Ctrl+Alt+T) and run the following commands:
... then post the outputs here.
sudo parted -l
sudo fdisk -l
"Evolution is Nature's way of issuing upgrades."