Solve y=-1/2left(x-4right)^2+6

$$y = - \frac{ 1 }{ 2 } { \left(x-4 \right) }^{ 2 } +6$$

Solve for y

$$y=-\frac{\left(x-4\right)^{2}}{2}+6$$

Solution Steps

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

$$y=-\frac{1}{2}\left(x^{2}-8x+16\right)+6$$

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

$$y=-\frac{1}{2}x^{2}+4x-8+6$$

Add -8 and 6 to get -2.

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

Solve for x (complex solution)

$$x=-\sqrt{12-2y}+4$$

$$x=\sqrt{12-2y}+4$$

Solve for x

$$x=-\sqrt{12-2y}+4$$

$$x=\sqrt{12-2y}+4\text{, }y\leq 6$$