Solve 3/x+2+2/n=4x-4/x^2-4

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

Solve for n (complex solution)

$$n=2\left(x-2\right)$$

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

Steps for Solving Linear Equation

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n\left(x-2\right)\left(x+2\right), the least common multiple of x+2,n,x^{2}-4.

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

Use the distributive property to multiply n by x-2.

$$\left(nx-2n\right)\times 3+\left(x^{2}-4\right)\times 2=n\left(4x-4\right)$$

Use the distributive property to multiply nx-2n by 3.

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

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

$$3nx-6n+2x^{2}-8=n\left(4x-4\right)$$

Use the distributive property to multiply n by 4x-4.

$$3nx-6n+2x^{2}-8=4nx-4n$$

Subtract 4nx from both sides.

$$3nx-6n+2x^{2}-8-4nx=-4n$$

Combine 3nx and -4nx to get -nx.

$$-nx-6n+2x^{2}-8=-4n$$

Add 4n to both sides.

$$-nx-6n+2x^{2}-8+4n=0$$

Combine -6n and 4n to get -2n.

$$-nx-2n+2x^{2}-8=0$$

Subtract 2x^{2} from both sides. Anything subtracted from zero gives its negation.

$$-nx-2n-8=-2x^{2}$$

Add 8 to both sides.

$$-nx-2n=-2x^{2}+8$$

Combine all terms containing n.

$$\left(-x-2\right)n=-2x^{2}+8$$

The equation is in standard form.

$$\left(-x-2\right)n=8-2x^{2}$$

Divide both sides by -x-2.

$$\frac{\left(-x-2\right)n}{-x-2}=\frac{8-2x^{2}}{-x-2}$$

Dividing by -x-2 undoes the multiplication by -x-2.

$$n=\frac{8-2x^{2}}{-x-2}$$

Divide -2x^{2}+8 by -x-2.

$$n=2x-4$$

Variable n cannot be equal to 0.

$$n=2x-4\text{, }n\neq 0$$

Solve for n

$$n=2\left(x-2\right)$$

$$|x|\neq 2$$

Steps for Solving Linear Equation

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n\left(x-2\right)\left(x+2\right), the least common multiple of x+2,n,x^{2}-4.

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

Use the distributive property to multiply n by x-2.

$$\left(nx-2n\right)\times 3+\left(x^{2}-4\right)\times 2=n\left(4x-4\right)$$

Use the distributive property to multiply nx-2n by 3.

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

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

$$3nx-6n+2x^{2}-8=n\left(4x-4\right)$$

Use the distributive property to multiply n by 4x-4.

$$3nx-6n+2x^{2}-8=4nx-4n$$

Subtract 4nx from both sides.

$$3nx-6n+2x^{2}-8-4nx=-4n$$

Combine 3nx and -4nx to get -nx.

$$-nx-6n+2x^{2}-8=-4n$$

Add 4n to both sides.

$$-nx-6n+2x^{2}-8+4n=0$$

Combine -6n and 4n to get -2n.

$$-nx-2n+2x^{2}-8=0$$

Subtract 2x^{2} from both sides. Anything subtracted from zero gives its negation.

$$-nx-2n-8=-2x^{2}$$

Add 8 to both sides.

$$-nx-2n=-2x^{2}+8$$

Combine all terms containing n.

$$\left(-x-2\right)n=-2x^{2}+8$$

The equation is in standard form.

$$\left(-x-2\right)n=8-2x^{2}$$

Divide both sides by -x-2.

$$\frac{\left(-x-2\right)n}{-x-2}=\frac{8-2x^{2}}{-x-2}$$

Dividing by -x-2 undoes the multiplication by -x-2.

$$n=\frac{8-2x^{2}}{-x-2}$$

Divide -2x^{2}+8 by -x-2.

$$n=2x-4$$

Variable n cannot be equal to 0.

$$n=2x-4\text{, }n\neq 0$$

Solve for x

$$x=\frac{n+4}{2}$$

$$n\neq -8\text{ and }n\neq 0$$