I think you need a further condition otherwise

a=-2x

b=x

c=x

satisfies your equation for all x.

If you have the condition that a,b,c > 0 then what you can do is let

$a=x+\delta$

$b=x$

$c=x$

and solve for $\delta$ showing it must be equal to zero.

Then by symmetry of the problem the same result applies to $b$ and $c$.