Solve z^2+(z-z_{1})z+7-4i=0

$$z ^ { 2 } + ( z - z _ { 1 } ) z + 7 - 4 i = 0$$

Solve for z_1

$$z_{1}=2z+\frac{7-4i}{z}$$

$$z\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply z-z_{1} by z.

$$z^{2}+z^{2}-z_{1}z+7-4i=0$$

Combine z^{2} and z^{2} to get 2z^{2}.

$$2z^{2}-z_{1}z+7-4i=0$$

Subtract 2z^{2} from both sides. Anything subtracted from zero gives its negation.

$$-z_{1}z+7-4i=-2z^{2}$$

Subtract 7 from both sides.

$$-z_{1}z-4i=-2z^{2}-7$$

Add 4i to both sides.

$$-z_{1}z=-2z^{2}-7+4i$$

The equation is in standard form.

$$\left(-z\right)z_{1}=-7+4i-2z^{2}$$

Divide both sides by -z.

$$\frac{\left(-z\right)z_{1}}{-z}=\frac{-7+4i-2z^{2}}{-z}$$

Dividing by -z undoes the multiplication by -z.

$$z_{1}=\frac{-7+4i-2z^{2}}{-z}$$

Divide -2z^{2}+\left(-7+4i\right) by -z.

$$z_{1}=2z+\frac{7-4i}{z}$$

Solve for z

$$z=\frac{\sqrt{z_{1}^{2}+\left(-56+32i\right)}+z_{1}}{4}$$

$$z=\frac{-\sqrt{z_{1}^{2}+\left(-56+32i\right)}+z_{1}}{4}$$