These are the instructions to root Samsung Galaxy Grand Quattro I8552 (running on android 4.2.2) using Framaroot Application on Phone itself (does not require computer).
Requirements: Your Samsung Galaxy Grand Quattro I8552 should have atleast 10-20 percent of battery to perform the rooting process.
Step 1: Enable USB Debugging on your Galaxy Grand Quattro I8552. To enable USB Debugging: Open Settings > Developers Option > USB Debugging > Tick to enable.
(Developers Option is hidden for Android 4.2.2 or above, to unhide Go to Settings > About Phone > Built Number (Tap on it for 5-8 times).
Step 2: On your Device, Open Settings > Security > Device Administration >Unknown Sources (tick to enable it).
Step 3: Download (latest) and install Framaroot Application on your Device. Once installation is completed you will be able to see the Framaroot Icon in your Menu.
Step 4: Now, Tap on the Framaroot App Icon to Open it.
Step 5: Once Framaroot Application is launched you will be able to see the following screen:
Step 6: In Framaroot, You will be able to see 2 options i.e Aragorn and Galdalf. Tap on the Galdalf Option.
Step 7: Now, you will be able to see the success popup on the Phone Screen.
Step 8: Now, you have to restart your Device.
Step 9: Now your Samsung Galaxy Grand Quattro I8552 is Rooted. To check whether your phone is rooted correctly, open Applications Menu, there you will be able to see a new app installed, called SuperSU. If this app exist then it means you have successfully rooted your device.
Optional: You can also verify the root status of your device by using the Root Checker Application.
[*] No data loss process: Your Samsung Galaxy Grand Quattro I8552 Phone Data like contacts, pictures, videos, music and etc will be not affected with this method.
[*] Device Warranty: Once you full unroot your device through SuperSU, you can claim for the phone warranty. In short no need to worry for phone warranty.
[*] Credits: Framaroot Application is developed by alephzain (XDA Developer).
[*] If you want to install CWM recovery on Galaxy Grand Quattro then see How to root and install Grand Quattro (another method to root).