Solving $ \sqrt{x - 4} + \sqrt{x - 7} = 1 $. I have the equation $ \sqrt{x - 4} + \sqrt{x - 7} = 1 $.
I tried to square both sides, but then I got a more difficult equation:
$$
2 x - 11 + 2 \sqrt{x^{2} - 11 x - 28} = 1.
$$
Can someone tell me what I should do next?

Suppose that $x$ is a solution. Flip things over. We get
$$\frac{1}{\sqrt{x-4}+\sqrt{x-7}}=1.$$
Rationalizing the denominator, and minor manipulation gives
$$\sqrt{x-4}-\sqrt{x-7}=3.$$
From this, addition gives $2\sqrt{x-4}=4$. The only possibility is $x=8$, which is not a solution.  
