Every Problem Has A Solution, isn't it?

EDIT: this solution is not correct, see Nikola Alfredi's solution for correction!

$\sqrt{(x-1)(x-3)}$ + $\sqrt{(x+3)(x-3)}$ = $\sqrt{2(2x-1)(x-3)}$

Notice that $\sqrt{x-3}$ is a common factor in each of these terms.

$\sqrt{x-3}$ $[\sqrt{x-1}$ + $\sqrt{x+3}]$ = $\sqrt{x-3}$ $\sqrt{2(2x-1)}$

Factoring that out, we can eliminate it from both sides of the equation:

$\sqrt{x-1}$ + $\sqrt{x+3}$ = $\sqrt{2(2x-1)}$

Now, square both sides to get rid of the radicals:

$(x-1)$ + $(x+3)$ = $2(2x-1)$

Finally, simplify the above equation to solve for x:

$2x+2=4x-2$

$4=2x$

$2=x$

Therefore, there is only one solution to the equation: $x=2$