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

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

Solve for y

$$y=-\left(x-1\right)^{2}+3$$

Solution Steps

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

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

To find the opposite of x^{2}-2x+1, find the opposite of each term.

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

Subtract 1 from 2 to get 1.

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

Add 1 to both sides.

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

Add 1 and 1 to get 2.

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

Solve for x (complex solution)

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

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

Solve for x

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

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