Edit: Corsair has a CX 750w for around $80 but I don't recommend the CX series for heavy load systems like crossfire mainly due to their cheap capacitors (CX series is a CWT brand PSU, chinese made OEM)
I think possibly not.
You need 2 6 pin pcie connectors for each card.
A Corsair CX600(if that is what you have) only has two.
Normally a psu will have all the connectors that it has the power to support.
But... the cx600 has 2 6+2 connectors which indicates that it just might have enough.
You could try it, and be prepared to shell out $80-$100 for a quality 700w+ psu.
Or, consider the option of marketing your R9-270X and replacing it with a single great card.
Your psu can support any single gpu card made today.
Plus, you would avoid any possible cf issues like stuttering or tearing.