Solve y=9(x+1/2)^2-13/4

$$y = 9 ( x + \frac { 1 } { 2 } ) ^ { 2 } - \frac { 13 } { 4 }$$

Solve for y (complex solution)

$$y=9x^{2}+9x-1$$

Solution Steps

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

$$y=9\left(x^{2}+x+\frac{1}{4}\right)-\frac{13}{4}$$

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

$$y=9x^{2}+9x+\frac{9}{4}-\frac{13}{4}$$

Subtract \frac{13}{4} from \frac{9}{4} to get -1.

$$y=9x^{2}+9x-1$$

Solve for y

$$y=\frac{9\left(2x+1\right)^{2}-13}{4}$$

Solution Steps

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

$$y=9\left(x^{2}+x+\frac{1}{4}\right)-\frac{13}{4}$$

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

$$y=9x^{2}+9x+\frac{9}{4}-\frac{13}{4}$$

Subtract \frac{13}{4} from \frac{9}{4} to get -1.

$$y=9x^{2}+9x-1$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{4y+13}}{6}-\frac{1}{2}$$

$$x=\frac{\sqrt{4y+13}}{6}-\frac{1}{2}$$

Solve for x

$$x=-\frac{\sqrt{4y+13}}{6}-\frac{1}{2}$$

$$x=\frac{\sqrt{4y+13}}{6}-\frac{1}{2}\text{, }y\geq -\frac{13}{4}$$