Solve y={left(x-3/2right)}^2-1/4

$$y= { \left(x- \frac{ 3 }{ 2 } \right) }^{ 2 } - \frac{ 1 }{ 4 }$$

Solve for y

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

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{3}{2}\right)^{2}.

$$y=x^{2}-3x+\frac{9}{4}-\frac{1}{4}$$

Subtract \frac{1}{4} from \frac{9}{4} to get 2.

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

Solve for x (complex solution)

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

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

Solve for x

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

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