How to securely and completely wipe my HDD and SSD?

It depends on how paranoid you are. It's nearly impossible to recover data from a drive that's in pieces. If you don't want to destroy hardware:

1. Write random data to hard drives.

2. For SSDs, many support TCG OPAL. If so, rekey the drive. I've used this software[0] and this command[1]. If your drive doesn't support OPAL, you're on your own. Due to reallocation and over-provisioning, it's impossible to tell if your data is still on the drive even if you've completely overwritten it.

[0] https://github.com/Drive-Trust-Alliance/sedutil

[1] https://github.com/Drive-Trust-Alliance/sedutil/wiki/PSID-Re...

You said bulletproof, so the only two options I know are:

1) Physical Drive Shredder. You can pay companies to do this for you if you don't know anyone that owns one.

2) Kiln. You can melt the drive. Avoid inhaling the fumes. Some regions may consider this a violation of environmental laws, so look that one up first.

Any form of software wiping or changing encryption keys leaves doubt. Data, blocks, keys could in theory be backed up in protected sectors that require vendor tools to view or change.

The above are just to answer the question of "bulletproof". They are not really the answer in your case however. You are selling the computer, so just take the drive out of it and put in another used one. You can find used drives online. Or just say, "No HD included"

Hammer.

Also you probably get much more direct and better answers by using a search engine. This is not a unique question and is trivially answered with some research.

The short answer is that a regular spinning HD can be wiped with bcwipe or any other equivalent, but the SSD really can't be wiped in any secure manner that is trustworthy.