We all know that E=mc².

Hence c = (+/-) sq.root (E/m)

 

What is the significance of the negative root of this equation?

And how does it reconcile with our understanding of c?

There's no significance to it. E=mc² is the statement that mass is a form of energy (that can be converted to other forms of energy), and that the proportionality factor between units of mass and that of energy is c². It's not a mystical entry point to the understanding of the nature of physical processes (although admittedly knowing that mass can be converted to other forms of energy is required for certain processes to make sense).

What timo said: there is no significance. c isn't defined in terms of E/m, it's defined as a constant. There's no point in rearranging the equation to get this.