Little question

Also some different brands have compatibility issues, I highly recommend taking the exact same sticks to expand. Not saying it will not work, but there's also no guarantee it will.

 

Back on my decade old pentium 4, I stuck in 2 512MB sticks onto an existing 512MB. Normal activities went well but when I opened a game I got blue screened. I just removed the old one and everything was fine.

 

Man that was a really old PC I was using up till 2012. Pretty much all RAM was being used constantly. The sticks worked independently just not together.

 

Your correct with the answer you found, you'll have to run all your memory at 1333MHz, using the 1333MHz memory's RAM timings and CAS latency. In short; Will it work? Yes. Will be faster? No, it'll likely be noticeably slower.

Although I do need to correct you; blue screens most certainly do not mean the memory is 'broke'. There are numerous different blue screens, they are very useful for identifying what part of a system is unstable. Unstable in this context, meaning, e.g., your CPU is being undervolted, or your RAM doesn't support the clock speed it's been set. You'll encounter blue screens all the time when you test an overclock, they'll become your best friend.