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

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

Solve for y

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

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$y=-\frac{1}{2}\left(x^{2}-8x+16\right)-\frac{3}{2}$$

Use the distributive property to multiply -\frac{1}{2} by x^{2}-8x+16.

$$y=-\frac{1}{2}x^{2}+4x-8-\frac{3}{2}$$

Subtract \frac{3}{2} from -8 to get -\frac{19}{2}.

$$y=-\frac{1}{2}x^{2}+4x-\frac{19}{2}$$

Solve for x (complex solution)

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

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

Solve for x

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

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