Solve frac{3y_{0}}{x^2+3}=-frac{6y_{2}}{x^2-3}

$$\frac { 3 y _ { 0 } } { x ^ { 2 } + 3 } = - \frac { 6 y _ { 2 } } { x ^ { 2 } - 3 }$$

Solve for y_0 (complex solution)

$$y_{0}=-\frac{2y_{2}\left(x^{2}+3\right)}{x^{2}-3}$$

$$x\neq -\sqrt{3}i\text{ and }x\neq \sqrt{3}i\text{ and }x\neq -\sqrt{3}\text{ and }x\neq \sqrt{3}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x^{2}-3\right)\left(x^{2}+3\right), the least common multiple of x^{2}+3,x^{2}-3.

$$\left(x^{2}-3\right)\times 3y_{0}=-\left(x^{2}+3\right)\times 6y_{2}$$

Use the distributive property to multiply x^{2}-3 by 3.

$$\left(3x^{2}-9\right)y_{0}=-\left(x^{2}+3\right)\times 6y_{2}$$

Use the distributive property to multiply 3x^{2}-9 by y_{0}.

$$3x^{2}y_{0}-9y_{0}=-\left(x^{2}+3\right)\times 6y_{2}$$

Use the distributive property to multiply x^{2}+3 by 6.

$$3x^{2}y_{0}-9y_{0}=-\left(6x^{2}+18\right)y_{2}$$

Use the distributive property to multiply 6x^{2}+18 by y_{2}.

$$3x^{2}y_{0}-9y_{0}=-\left(6x^{2}y_{2}+18y_{2}\right)$$

To find the opposite of 6x^{2}y_{2}+18y_{2}, find the opposite of each term.

$$3x^{2}y_{0}-9y_{0}=-6x^{2}y_{2}-18y_{2}$$

Combine all terms containing y_{0}.

$$\left(3x^{2}-9\right)y_{0}=-6x^{2}y_{2}-18y_{2}$$

The equation is in standard form.

$$\left(3x^{2}-9\right)y_{0}=-6y_{2}x^{2}-18y_{2}$$

Divide both sides by 3x^{2}-9.

$$\frac{\left(3x^{2}-9\right)y_{0}}{3x^{2}-9}=-\frac{6y_{2}\left(x^{2}+3\right)}{3x^{2}-9}$$

Dividing by 3x^{2}-9 undoes the multiplication by 3x^{2}-9.

$$y_{0}=-\frac{6y_{2}\left(x^{2}+3\right)}{3x^{2}-9}$$

Divide -6y_{2}\left(x^{2}+3\right) by 3x^{2}-9.

$$y_{0}=-\frac{2y_{2}\left(x^{2}+3\right)}{x^{2}-3}$$

Solve for y_0

$$y_{0}=-\frac{2y_{2}\left(x^{2}+3\right)}{x^{2}-3}$$

$$|x|\neq \sqrt{3}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x^{2}-3\right)\left(x^{2}+3\right), the least common multiple of x^{2}+3,x^{2}-3.

$$\left(x^{2}-3\right)\times 3y_{0}=-\left(x^{2}+3\right)\times 6y_{2}$$

Use the distributive property to multiply x^{2}-3 by 3.

$$\left(3x^{2}-9\right)y_{0}=-\left(x^{2}+3\right)\times 6y_{2}$$

Use the distributive property to multiply 3x^{2}-9 by y_{0}.

$$3x^{2}y_{0}-9y_{0}=-\left(x^{2}+3\right)\times 6y_{2}$$

Use the distributive property to multiply x^{2}+3 by 6.

$$3x^{2}y_{0}-9y_{0}=-\left(6x^{2}+18\right)y_{2}$$

Use the distributive property to multiply 6x^{2}+18 by y_{2}.

$$3x^{2}y_{0}-9y_{0}=-\left(6x^{2}y_{2}+18y_{2}\right)$$

To find the opposite of 6x^{2}y_{2}+18y_{2}, find the opposite of each term.

$$3x^{2}y_{0}-9y_{0}=-6x^{2}y_{2}-18y_{2}$$

Combine all terms containing y_{0}.

$$\left(3x^{2}-9\right)y_{0}=-6x^{2}y_{2}-18y_{2}$$

The equation is in standard form.

$$\left(3x^{2}-9\right)y_{0}=-6y_{2}x^{2}-18y_{2}$$

Divide both sides by 3x^{2}-9.

$$\frac{\left(3x^{2}-9\right)y_{0}}{3x^{2}-9}=-\frac{6y_{2}\left(x^{2}+3\right)}{3x^{2}-9}$$

Dividing by 3x^{2}-9 undoes the multiplication by 3x^{2}-9.

$$y_{0}=-\frac{6y_{2}\left(x^{2}+3\right)}{3x^{2}-9}$$

Divide -6y_{2}\left(x^{2}+3\right) by 3x^{2}-9.

$$y_{0}=-\frac{2y_{2}\left(x^{2}+3\right)}{x^{2}-3}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-i\left(y_{0}+2y_{2}\right)^{-\frac{1}{2}}\sqrt{6y_{2}-3y_{0}}\text{; }x=i\left(y_{0}+2y_{2}\right)^{-\frac{1}{2}}\sqrt{6y_{2}-3y_{0}}\text{, }&y_{0}\neq 0\text{ and }y_{0}\neq -2y_{2}\text{ and }y_{2}\neq 0\\x\in \mathrm{C}\setminus -\sqrt{3}i,\sqrt{3}i,-\sqrt{3},\sqrt{3}\text{, }&y_{0}=0\text{ and }y_{2}=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\sqrt{-\frac{3\left(2y_{2}-y_{0}\right)}{y_{0}+2y_{2}}}\text{; }x=-\sqrt{-\frac{3\left(2y_{2}-y_{0}\right)}{y_{0}+2y_{2}}}\text{, }&\left(y_{2}\neq 0\text{ and }|y_{0}|\geq 2|y_{2}|\text{ and }y_{0}\geq 2y_{2}\text{ and }y_{0}>-2y_{2}\right)\text{ or }\left(y_{2}\neq 0\text{ and }|y_{0}|\geq 2|y_{2}|\text{ and }y_{0}\leq 2y_{2}\text{ and }y_{0}<-2y_{2}\right)\text{ or }\left(y_{2}\neq 0\text{ and }y_{0}=2y_{2}\right)\\x\in \mathrm{R}\setminus \sqrt{3},-\sqrt{3}\text{, }&y_{0}=0\text{ and }y_{2}=0\end{matrix}\right.$$