How to factor $x + 1 - 2 \sqrt x$?

Question

My teacher said the answer is $(\sqrt x -1)^2$, but I want to know how he figured it out. I know it's a trick I learned years ago, but I can't remember how to do this.

Answer 1

Substitute $y:=\sqrt x$ to get $y^2-2y+1$

Answer 2

This is a quadratic in $\sqrt{x}$. If you let $\sqrt{x} = y$, this becomes $y^2 - 2y - 1$ which can be factorized.

Answer 3

$$\begin{align} x + 1 − 2\sqrt{x}&=x + 1 − \sqrt{x}- \sqrt{x}\\ &= x − \sqrt{x} + 1 - \sqrt{x}\\ &= \sqrt{x}(\sqrt{x} - 1) - 1(\sqrt{x} - 1)\\ &= (\sqrt{x} - 1)(\sqrt{x} - 1)\\ &= (\sqrt{x} - 1)^{2} \end{align} $$