Solve 3/x-2-2=3m/x^2-4+5

$$\frac { 3 } { x - 2 } - 2 = \frac { 3 m } { x ^ { 2 } - 4 } + 5$$

Solve for m (complex solution)

$$m=-\frac{7x^{2}}{3}+x+\frac{34}{3}$$

$$x\neq -2\text{ and }x\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x-2,x^{2}-4.

$$\left(x+2\right)\times 3+\left(x-2\right)\left(x+2\right)\left(-2\right)=3m+\left(x-2\right)\left(x+2\right)\times 5$$

Use the distributive property to multiply x+2 by 3.

$$3x+6+\left(x-2\right)\left(x+2\right)\left(-2\right)=3m+\left(x-2\right)\left(x+2\right)\times 5$$

Use the distributive property to multiply x-2 by x+2 and combine like terms.

$$3x+6+\left(x^{2}-4\right)\left(-2\right)=3m+\left(x-2\right)\left(x+2\right)\times 5$$

Use the distributive property to multiply x^{2}-4 by -2.

$$3x+6-2x^{2}+8=3m+\left(x-2\right)\left(x+2\right)\times 5$$

Add 6 and 8 to get 14.

$$3x+14-2x^{2}=3m+\left(x-2\right)\left(x+2\right)\times 5$$

Use the distributive property to multiply x-2 by x+2 and combine like terms.

$$3x+14-2x^{2}=3m+\left(x^{2}-4\right)\times 5$$

Use the distributive property to multiply x^{2}-4 by 5.

$$3x+14-2x^{2}=3m+5x^{2}-20$$

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

$$3m+5x^{2}-20=3x+14-2x^{2}$$

Subtract 5x^{2} from both sides.

$$3m-20=3x+14-2x^{2}-5x^{2}$$

Combine -2x^{2} and -5x^{2} to get -7x^{2}.

$$3m-20=3x+14-7x^{2}$$

Add 20 to both sides.

$$3m=3x+14-7x^{2}+20$$

Add 14 and 20 to get 34.

$$3m=3x+34-7x^{2}$$

The equation is in standard form.

$$3m=34+3x-7x^{2}$$

Divide both sides by 3.

$$\frac{3m}{3}=-\frac{\left(7x-17\right)\left(x+2\right)}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$m=-\frac{\left(7x-17\right)\left(x+2\right)}{3}$$

Divide -\left(-17+7x\right)\left(2+x\right) by 3.

$$m=-\frac{7x^{2}}{3}+x+\frac{34}{3}$$

Solve for m

$$m=-\frac{7x^{2}}{3}+x+\frac{34}{3}$$

$$|x|\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x-2,x^{2}-4.

$$\left(x+2\right)\times 3+\left(x-2\right)\left(x+2\right)\left(-2\right)=3m+\left(x-2\right)\left(x+2\right)\times 5$$

Use the distributive property to multiply x+2 by 3.

$$3x+6+\left(x-2\right)\left(x+2\right)\left(-2\right)=3m+\left(x-2\right)\left(x+2\right)\times 5$$

Use the distributive property to multiply x-2 by x+2 and combine like terms.

$$3x+6+\left(x^{2}-4\right)\left(-2\right)=3m+\left(x-2\right)\left(x+2\right)\times 5$$

Use the distributive property to multiply x^{2}-4 by -2.

$$3x+6-2x^{2}+8=3m+\left(x-2\right)\left(x+2\right)\times 5$$

Add 6 and 8 to get 14.

$$3x+14-2x^{2}=3m+\left(x-2\right)\left(x+2\right)\times 5$$

Use the distributive property to multiply x-2 by x+2 and combine like terms.

$$3x+14-2x^{2}=3m+\left(x^{2}-4\right)\times 5$$

Use the distributive property to multiply x^{2}-4 by 5.

$$3x+14-2x^{2}=3m+5x^{2}-20$$

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

$$3m+5x^{2}-20=3x+14-2x^{2}$$

Subtract 5x^{2} from both sides.

$$3m-20=3x+14-2x^{2}-5x^{2}$$

Combine -2x^{2} and -5x^{2} to get -7x^{2}.

$$3m-20=3x+14-7x^{2}$$

Add 20 to both sides.

$$3m=3x+14-7x^{2}+20$$

Add 14 and 20 to get 34.

$$3m=3x+34-7x^{2}$$

The equation is in standard form.

$$3m=34+3x-7x^{2}$$

Divide both sides by 3.

$$\frac{3m}{3}=-\frac{\left(7x-17\right)\left(x+2\right)}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$m=-\frac{\left(7x-17\right)\left(x+2\right)}{3}$$

Divide -\left(-17+7x\right)\left(2+x\right) by 3.

$$m=-\frac{7x^{2}}{3}+x+\frac{34}{3}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{961-84m}+3}{14}\text{, }&m\neq 4\\x=\frac{-\sqrt{961-84m}+3}{14}\text{, }&m\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{961-84m}+3}{14}\text{, }&m\neq 4\text{ and }m\leq \frac{961}{84}\\x=\frac{-\sqrt{961-84m}+3}{14}\text{, }&m\leq \frac{961}{84}\text{ and }m\neq 0\end{matrix}\right.$$