Solve x-1=y+(x-1)^2

$$x - 1 = y + ( x - 1 ) ^ { 2 }$$

Solve for y

$$y=\left(1-x\right)\left(x-2\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x-1=y+x^{2}-2x+1$$

Swap sides so that all variable terms are on the left hand side.

$$y+x^{2}-2x+1=x-1$$

Subtract x^{2} from both sides.

$$y-2x+1=x-1-x^{2}$$

Add 2x to both sides.

$$y+1=x-1-x^{2}+2x$$

Combine x and 2x to get 3x.

$$y+1=3x-1-x^{2}$$

Subtract 1 from both sides.

$$y=3x-1-x^{2}-1$$

Subtract 1 from -1 to get -2.

$$y=3x-2-x^{2}$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{1-4y}+3}{2}$$

$$x=\frac{\sqrt{1-4y}+3}{2}$$

Solve for x

$$x=\frac{\sqrt{1-4y}+3}{2}$$

$$x=\frac{-\sqrt{1-4y}+3}{2}\text{, }y\leq \frac{1}{4}$$