# How do you solve x+1=(x+1)^2?

Jul 8, 2016

$x \in \left\{- 1 , 0\right\}$

#### Explanation:

We could expand the right hand side, gather all the terms on one side, and then solve the resulting quadratic, however because everything is currently in terms of $x + 1$, let's use another approach.

Let $u = x + 1$

Then we have

$u = {u}^{2}$

$\implies {u}^{2} - u = 0$

$\implies u \left(u - 1\right) = 0$

$\implies u = 0 \mathmr{and} u - 1 = 0$

$\implies x + 1 = 0 \mathmr{and} x + 1 - 1 = 0$

$\therefore x = - 1 \mathmr{and} x = 0$