Solve x/x^2-4x+4-8/x^2-2x=1/x

$$\frac { x } { x ^ { 2 } - 4 x + 4 } - \frac { 8 } { x ^ { 2 } - 2 x } = \frac { 1 } { x }$$

Solve for x

$$x=3$$

Steps for Solving Linear Equation

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

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

Multiply x and x to get x^{2}.

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

Use the distributive property to multiply x-2 by 8.

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

To find the opposite of 8x-16, find the opposite of each term.

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

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

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

Subtract x^{2} from both sides.

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

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

$$-8x+16=-4x+4$$

Add 4x to both sides.

$$-8x+16+4x=4$$

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

$$-4x+16=4$$

Subtract 16 from both sides.

$$-4x=4-16$$

Subtract 16 from 4 to get -12.

$$-4x=-12$$

Divide both sides by -4.

$$x=\frac{-12}{-4}$$

Divide -12 by -4 to get 3.

$$x=3$$