Solve 18x_{2}+10x_{4}+6x_{1}x_{2}-12-2w^2=0

$$18 x _ { 2 } + 10 x _ { 4 } + 6 x _ { 1 } x _ { 2 } - 12 - 2 w ^ { 2 } = 0$$

Solve for x_1 (complex solution)

$$\left\{\begin{matrix}x_{1}=-\frac{9x_{2}+5x_{4}-w^{2}-6}{3x_{2}}\text{, }&x_{2}\neq 0\\x_{1}\in \mathrm{C}\text{, }&x_{2}=0\text{ and }x_{4}=\frac{w^{2}+6}{5}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 18x_{2} from both sides. Anything subtracted from zero gives its negation.

$$10x_{4}+6x_{1}x_{2}-12-2w^{2}=-18x_{2}$$

Subtract 10x_{4} from both sides.

$$6x_{1}x_{2}-12-2w^{2}=-18x_{2}-10x_{4}$$

Add 12 to both sides.

$$6x_{1}x_{2}-2w^{2}=-18x_{2}-10x_{4}+12$$

Add 2w^{2} to both sides.

$$6x_{1}x_{2}=-18x_{2}-10x_{4}+12+2w^{2}$$

The equation is in standard form.

$$6x_{2}x_{1}=12+2w^{2}-10x_{4}-18x_{2}$$

Divide both sides by 6x_{2}.

$$\frac{6x_{2}x_{1}}{6x_{2}}=\frac{12+2w^{2}-10x_{4}-18x_{2}}{6x_{2}}$$

Dividing by 6x_{2} undoes the multiplication by 6x_{2}.

$$x_{1}=\frac{12+2w^{2}-10x_{4}-18x_{2}}{6x_{2}}$$

Divide -18x_{2}-10x_{4}+12+2w^{2} by 6x_{2}.

$$x_{1}=\frac{6+w^{2}-5x_{4}-9x_{2}}{3x_{2}}$$

Solve for x_1

$$\left\{\begin{matrix}x_{1}=-\frac{9x_{2}+5x_{4}-w^{2}-6}{3x_{2}}\text{, }&x_{2}\neq 0\\x_{1}\in \mathrm{R}\text{, }&x_{2}=0\text{ and }x_{4}=\frac{w^{2}+6}{5}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 18x_{2} from both sides. Anything subtracted from zero gives its negation.

$$10x_{4}+6x_{1}x_{2}-12-2w^{2}=-18x_{2}$$

Subtract 10x_{4} from both sides.

$$6x_{1}x_{2}-12-2w^{2}=-18x_{2}-10x_{4}$$

Add 12 to both sides.

$$6x_{1}x_{2}-2w^{2}=-18x_{2}-10x_{4}+12$$

Add 2w^{2} to both sides.

$$6x_{1}x_{2}=-18x_{2}-10x_{4}+12+2w^{2}$$

The equation is in standard form.

$$6x_{2}x_{1}=12+2w^{2}-10x_{4}-18x_{2}$$

Divide both sides by 6x_{2}.

$$\frac{6x_{2}x_{1}}{6x_{2}}=\frac{12+2w^{2}-10x_{4}-18x_{2}}{6x_{2}}$$

Dividing by 6x_{2} undoes the multiplication by 6x_{2}.

$$x_{1}=\frac{12+2w^{2}-10x_{4}-18x_{2}}{6x_{2}}$$

Divide -18x_{2}-10x_{4}+12+2w^{2} by 6x_{2}.

$$x_{1}=\frac{6+w^{2}-5x_{4}-9x_{2}}{3x_{2}}$$

Solve for w (complex solution)

$$w=-\sqrt{3x_{1}x_{2}+9x_{2}+5x_{4}-6}$$

$$w=\sqrt{3x_{1}x_{2}+9x_{2}+5x_{4}-6}$$

Solve for w

$$w=\sqrt{3x_{1}x_{2}+9x_{2}+5x_{4}-6}$$

$$w=-\sqrt{3x_{1}x_{2}+9x_{2}+5x_{4}-6}\text{, }x_{4}\geq \frac{6-9x_{2}-3x_{1}x_{2}}{5}$$