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

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

Solve for y

$$y=2x^{2}-2x+5$$

Solution Steps

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

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

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

$$y=x^{2}+2x+1+x^{2}-4x+4$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$y=2x^{2}+2x+1-4x+4$$

Combine 2x and -4x to get -2x.

$$y=2x^{2}-2x+1+4$$

Add 1 and 4 to get 5.

$$y=2x^{2}-2x+5$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{2y-9}+1}{2}$$

$$x=\frac{\sqrt{2y-9}+1}{2}$$

Solve for x

$$x=\frac{-\sqrt{2y-9}+1}{2}$$

$$x=\frac{\sqrt{2y-9}+1}{2}\text{, }y\geq \frac{9}{2}$$