Solve y=-((x+3/2)^2+5-9/4)

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

Solve for y

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

Solution Steps

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

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

Add \frac{9}{4} and 5 to get \frac{29}{4}.

$$y=-\left(x^{2}+3x+\frac{29}{4}-\frac{9}{4}\right)$$

Subtract \frac{9}{4} from \frac{29}{4} to get 5.

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

To find the opposite of x^{2}+3x+5, find the opposite of each term.

$$y=-x^{2}-3x-5$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{-4y-11}-3}{2}$$

$$x=\frac{\sqrt{-4y-11}-3}{2}$$

Solve for x

$$x=\frac{-\sqrt{-4y-11}-3}{2}$$

$$x=\frac{\sqrt{-4y-11}-3}{2}\text{, }y\leq -\frac{11}{4}$$