<s>[INST] Find x for $\sqrt{(5x-1)}+\sqrt{(x-1)}=2$ Solve:
$$\sqrt{(5x-1)}+\sqrt{(x-1)}=2$$
When $x=1$, we get the following equation to equal to $2$
I've been trying to solve this problem but when I square both sides and simplify I end up with:
$$x^2+6x+2=0$$ and of course $x=1$ cannot be a solution. So im not sure what im doing wrong. Any help on this problem?
 [/INST] Perhaps a slightly tidier solution:
$$\begin{align}
\sqrt{5x-1}+\sqrt{x-1}&=2 \\
\sqrt{5x-1} &= 2-\sqrt{x-1} &\text{from regrouping}\\
5x-1&=4-4\sqrt{x-1}+(x-1) & \text{after squaring}\\
4x-4&=4\sqrt{x-1} &\text{after regrouping again}\\
(x-1)^2&=x-1 &\text{divide by 4 and square again}\\
(x-1)^2-(x-1)&=0 \\
(x-1)(x)&=0 &\text{factor}
\end{align}$$
so either $x-1=0$ or $x=0$. Plugging back into the original equation we see that $x=1$ is a solution and $x=0$ is not.
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