Solve x=1/2(z^2+z)

$$x= \frac{ 1 }{ 2 } ( { z }^{ 2 } +z)$$

Solve for x

$$x=\frac{z\left(z+1\right)}{2}$$

Solution Steps

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

$$x=\frac{1}{2}z^{2}+\frac{1}{2}z$$

Solve for z (complex solution)

$$z=\frac{\sqrt{8x+1}-1}{2}$$

$$z=\frac{-\sqrt{8x+1}-1}{2}$$

Solve for z

$$z=\frac{\sqrt{8x+1}-1}{2}$$

$$z=\frac{-\sqrt{8x+1}-1}{2}\text{, }x\geq -\frac{1}{8}$$