Solve -2/2x^2-8+1/x=1/(x-2)^2+1/x+2

$$\frac { - 2 } { 2 x ^ { 2 } - 8 } + \frac { 1 } { x } = \frac { 1 } { ( x - 2 ) ^ { 2 } } + \frac { 1 } { x + 2 }$$

Solve for x

$$x=1$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -2,0,2 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+2\right)\left(x-2\right)^{2}, the least common multiple of 2x^{2}-8,x,\left(x-2\right)^{2},x+2.

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

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

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

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

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

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

Use the distributive property to multiply 2x+4 by x^{2}-4x+4 and combine like terms.

$$-2x^{2}+4x+2x^{3}-4x^{2}-8x+16=2\left(x+2\right)x+2x\left(x-2\right)^{2}$$

Combine -2x^{2} and -4x^{2} to get -6x^{2}.

$$-6x^{2}+4x+2x^{3}-8x+16=2\left(x+2\right)x+2x\left(x-2\right)^{2}$$

Combine 4x and -8x to get -4x.

$$-6x^{2}-4x+2x^{3}+16=2\left(x+2\right)x+2x\left(x-2\right)^{2}$$

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

$$-6x^{2}-4x+2x^{3}+16=\left(2x+4\right)x+2x\left(x-2\right)^{2}$$

Use the distributive property to multiply 2x+4 by x.

$$-6x^{2}-4x+2x^{3}+16=2x^{2}+4x+2x\left(x-2\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$-6x^{2}-4x+2x^{3}+16=2x^{2}+4x+2x\left(x^{2}-4x+4\right)$$

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

$$-6x^{2}-4x+2x^{3}+16=2x^{2}+4x+2x^{3}-8x^{2}+8x$$

Combine 2x^{2} and -8x^{2} to get -6x^{2}.

$$-6x^{2}-4x+2x^{3}+16=-6x^{2}+4x+2x^{3}+8x$$

Combine 4x and 8x to get 12x.

$$-6x^{2}-4x+2x^{3}+16=-6x^{2}+12x+2x^{3}$$

Add 6x^{2} to both sides.

$$-6x^{2}-4x+2x^{3}+16+6x^{2}=12x+2x^{3}$$

Combine -6x^{2} and 6x^{2} to get 0.

$$-4x+2x^{3}+16=12x+2x^{3}$$

Subtract 12x from both sides.

$$-4x+2x^{3}+16-12x=2x^{3}$$

Combine -4x and -12x to get -16x.

$$-16x+2x^{3}+16=2x^{3}$$

Subtract 2x^{3} from both sides.

$$-16x+2x^{3}+16-2x^{3}=0$$

Combine 2x^{3} and -2x^{3} to get 0.

$$-16x+16=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$-16x=-16$$

Divide both sides by -16.

$$x=\frac{-16}{-16}$$

Divide -16 by -16 to get 1.

$$x=1$$