Solve a^2+2a+c_{2}z=-3

$$a ^ { 2 } + 2 a + c _ { 2 } z = - 3$$

Solve for c_2 (complex solution)

$$\left\{\begin{matrix}c_{2}=-\frac{a^{2}+2a+3}{z}\text{, }&z\neq 0\\c_{2}\in \mathrm{C}\text{, }&\left(a=-\sqrt{2}i-1\text{ or }a=-1+\sqrt{2}i\right)\text{ and }z=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract a^{2} from both sides.

$$2a+c_{2}z=-3-a^{2}$$

Subtract 2a from both sides.

$$c_{2}z=-3-a^{2}-2a$$

The equation is in standard form.

$$zc_{2}=-a^{2}-2a-3$$

Divide both sides by z.

$$\frac{zc_{2}}{z}=\frac{-a^{2}-2a-3}{z}$$

Dividing by z undoes the multiplication by z.

$$c_{2}=\frac{-a^{2}-2a-3}{z}$$

Divide -3-a^{2}-2a by z.

$$c_{2}=-\frac{a^{2}+2a+3}{z}$$

Solve for c_2

$$c_{2}=-\frac{a^{2}+2a+3}{z}$$

$$z\neq 0$$

Steps for Solving Linear Equation

Subtract a^{2} from both sides.

$$2a+c_{2}z=-3-a^{2}$$

Subtract 2a from both sides.

$$c_{2}z=-3-a^{2}-2a$$

The equation is in standard form.

$$zc_{2}=-a^{2}-2a-3$$

Divide both sides by z.

$$\frac{zc_{2}}{z}=\frac{-a^{2}-2a-3}{z}$$

Dividing by z undoes the multiplication by z.

$$c_{2}=\frac{-a^{2}-2a-3}{z}$$

Divide -3-a^{2}-2a by z.

$$c_{2}=-\frac{a^{2}+2a+3}{z}$$

Solve for a (complex solution)

$$a=\sqrt{-c_{2}z-2}-1$$

$$a=-\sqrt{-c_{2}z-2}-1$$

Solve for a

$$a=\sqrt{-c_{2}z-2}-1$$

$$a=-\sqrt{-c_{2}z-2}-1\text{, }\left(z>0\text{ or }c_{2}\geq -\frac{2}{z}\right)\text{ and }\left(z<0\text{ or }c_{2}\leq -\frac{2}{z}\right)\text{ and }z\neq 0$$