Solve -1/2(x-5)^2+6=y

$$- \frac { 1 } { 2 } ( x - 5 ) ^ { 2 } + 6 = y$$

Solve for y

$$y=-\frac{\left(x-5\right)^{2}}{2}+6$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$-\frac{1}{2}\left(x^{2}-10x+25\right)+6=y$$

Use the distributive property to multiply -\frac{1}{2} by x^{2}-10x+25.

$$-\frac{1}{2}x^{2}+5x-\frac{25}{2}+6=y$$

Add -\frac{25}{2} and 6 to get -\frac{13}{2}.

$$-\frac{1}{2}x^{2}+5x-\frac{13}{2}=y$$

Swap sides so that all variable terms are on the left hand side.

$$y=-\frac{1}{2}x^{2}+5x-\frac{13}{2}$$

Solve for x (complex solution)

$$x=-\sqrt{12-2y}+5$$

$$x=\sqrt{12-2y}+5$$

Solve for x

$$x=-\sqrt{12-2y}+5$$

$$x=\sqrt{12-2y}+5\text{, }y\leq 6$$