Solve y=-2[-1/2(x-3)]^2+5

$$y = - 2 [ - \frac { 1 } { 2 } ( x - 3 ) ] ^ { 2 } + 5$$

Solve for y

$$y=-\frac{\left(x-3\right)^{2}}{2}+5$$

Solution Steps

Use the distributive property to multiply -\frac{1}{2} by x-3.

$$y=-2\left(-\frac{1}{2}x+\frac{3}{2}\right)^{2}+5$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-\frac{1}{2}x+\frac{3}{2}\right)^{2}.

$$y=-2\left(\frac{1}{4}x^{2}-\frac{3}{2}x+\frac{9}{4}\right)+5$$

Use the distributive property to multiply -2 by \frac{1}{4}x^{2}-\frac{3}{2}x+\frac{9}{4}.

$$y=-\frac{1}{2}x^{2}+3x-\frac{9}{2}+5$$

Add -\frac{9}{2} and 5 to get \frac{1}{2}.

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

Solve for x (complex solution)

$$x=\sqrt{10-2y}+3$$

$$x=-\sqrt{10-2y}+3$$

Solve for x

$$x=\sqrt{10-2y}+3$$

$$x=-\sqrt{10-2y}+3\text{, }y\leq 5$$