Solve 2/x^2-1-2/x^2+1=4/x^4-1

$$\frac { 2 } { x ^ { 2 } - 1 } - \frac { 2 } { x ^ { 2 } + 1 } = \frac { 4 } { x ^ { 4 } - 1 }$$

Solve for x (complex solution)

$$x\in \mathrm{C}\setminus -1,1,-i,i$$

Solution Steps

Variable x cannot be equal to any of the values -1,-i,i,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x-i\right)\left(x+i\right), the least common multiple of x^{2}-1,x^{2}+1,x^{4}-1.

$$\left(x^{2}+1\right)\times 2-\left(x^{2}-1\right)\times 2=4$$

Use the distributive property to multiply x^{2}+1 by 2.

$$2x^{2}+2-\left(x^{2}-1\right)\times 2=4$$

Use the distributive property to multiply x^{2}-1 by 2.

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

To find the opposite of 2x^{2}-2, find the opposite of each term.

$$2x^{2}+2-2x^{2}+2=4$$

Combine 2x^{2} and -2x^{2} to get 0.

$$2+2=4$$

Add 2 and 2 to get 4.

$$4=4$$

Compare 4 and 4.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Variable x cannot be equal to any of the values -i,i,-1,1.

$$x\in \mathrm{C}\setminus -i,i,-1,1$$

Solve for x

$$x\in \mathrm{R}\setminus 1,-1$$

Solution Steps

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x^{2}+1\right), the least common multiple of x^{2}-1,x^{2}+1,x^{4}-1.

$$\left(x^{2}+1\right)\times 2-\left(x^{2}-1\right)\times 2=4$$

Use the distributive property to multiply x^{2}+1 by 2.

$$2x^{2}+2-\left(x^{2}-1\right)\times 2=4$$

Use the distributive property to multiply x^{2}-1 by 2.

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

To find the opposite of 2x^{2}-2, find the opposite of each term.

$$2x^{2}+2-2x^{2}+2=4$$

Combine 2x^{2} and -2x^{2} to get 0.

$$2+2=4$$

Add 2 and 2 to get 4.

$$4=4$$

Compare 4 and 4.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Variable x cannot be equal to any of the values -1,1.

$$x\in \mathrm{R}\setminus -1,1$$