Sum of roots of an equation $\sqrt{x-1}+\sqrt{2x-1}=x$

Question

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.

You have divided by $x-1$, without considering the possibility $x=1$. — Macavity, Oct 16 '14 at 02:53
$x=5$ is correct. but you miss $x-1=0$, when you divide $\sqrt{x-1}$, you should be careful. — chenbai, Oct 16 '14 at 02:53

score 3 · Accepted Answer · answered Oct 16 '14 at 02:53

3

HINT:

$x=1$ is also a valid root where $x-1$ has been cancelled out assuming $x-1\ne0$

Observe that, though $0$ being additive identity, $x=0$ does not effect the sum is not a valid root

answered Oct 16 '14 at 02:53

lab bhattacharjee

score 0 · Answer 2 · answered Jul 30 '23 at 13:26

0

x=0 is an invalid root.

Given the initial equation $\sqrt{x-1}+\sqrt{2x-1}=x$, the domain for all the potential roots for $x$ is restricted to $x\ge1$.

However, $0<1$, meaning the root $x=0$ lies outside the domain.

answered Jul 30 '23 at 13:26

Ryan Wilson

2 Answers2