Solve w^2+2w+3=g(w)

$$w ^ { 2 } + 2 w + 3 = g ( w )$$

Solve for g

$$g=w+2+\frac{3}{w}$$

$$w\neq 0$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$gw=w^{2}+2w+3$$

The equation is in standard form.

$$wg=w^{2}+2w+3$$

Divide both sides by w.

$$\frac{wg}{w}=\frac{w^{2}+2w+3}{w}$$

Dividing by w undoes the multiplication by w.

$$g=\frac{w^{2}+2w+3}{w}$$

Divide w^{2}+2w+3 by w.

$$g=w+2+\frac{3}{w}$$

Solve for w (complex solution)

$$w=\frac{\sqrt{g^{2}-4g-8}+g-2}{2}$$

$$w=\frac{-\sqrt{g^{2}-4g-8}+g-2}{2}$$

Solve for w

$$w=\frac{\sqrt{g^{2}-4g-8}+g-2}{2}$$

$$w=\frac{-\sqrt{g^{2}-4g-8}+g-2}{2}\text{, }g\geq 2\sqrt{3}+2\text{ or }g\leq 2-2\sqrt{3}$$