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

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

Solve for x

$$x=4$$

Steps for Solving Linear Equation

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

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

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

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

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

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

Use the distributive property to multiply x^{2}-4x+4 by -1.

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

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

$$-2x+4x-4=4$$

Combine -2x and 4x to get 2x.

$$2x-4=4$$

Add 4 to both sides.

$$2x=4+4$$

Add 4 and 4 to get 8.

$$2x=8$$

Divide both sides by 2.

$$x=\frac{8}{2}$$

Divide 8 by 2 to get 4.

$$x=4$$