Sum of roots of an equation $\sqrt{x-1}+\sqrt{2x-1}=x$ Find the sum of the roots of the equation $\sqrt{x-1}+\sqrt{2x-1}=x$
My attempt: Squaring the equation: $(x-1)+(2x-1) +2\sqrt{(x-1)(2x-1)}=x^2$
$\implies x^2-3x+2=2\sqrt{(x-1)(2x-1)} $
$\implies (x-1)(x-2)=2\sqrt{(x-1)(2x-1)} $
$\implies (x-2)=2\sqrt{\displaystyle \frac{(2x-1)}{(x-1)}} $
Squaring, $(x^2-4x+4)(x-1)=8x-4$
$\implies x^2(x-5)=0$. So, the sum of roots should be five. 
The given answer is 6.
Could anyone look at my attempt to find where I went wrong. Thanks.

The problem is that you divided by $x-1$, so, you lost a root.Strating from $$ (x-1)(x-2)=2\sqrt{(x-1)(2x-1)}$$ as you properly wrote and squaring $$(x-1)^2(x-2)^2=4{(x-1)(2x-1)}$$ Expanding and grouping leads to $$x^4-6 x^3+5 x^2=0$$ so the sum of the roots is $6$ (you can check that the roots are $0,0,1,5$.
