Solve 1/x-1/x-2+a=0

$$\frac { 1 } { x } - \frac { 1 } { x - 2 } + a = 0$$

Solve for a

$$a=\frac{2}{x\left(x-2\right)}$$

$$x\neq 0\text{ and }x\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x,x-2.

$$x-2-x+x\left(x-2\right)a=0$$

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

$$x-2-x+\left(x^{2}-2x\right)a=0$$

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

$$x-2-x+x^{2}a-2xa=0$$

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

$$-2-x+x^{2}a-2xa=-x$$

Add 2 to both sides.

$$-x+x^{2}a-2xa=-x+2$$

Add x to both sides.

$$x^{2}a-2xa=-x+2+x$$

Combine -x and x to get 0.

$$x^{2}a-2xa=2$$

Combine all terms containing a.

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

Divide both sides by x^{2}-2x.

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

Dividing by x^{2}-2x undoes the multiplication by x^{2}-2x.

$$a=\frac{2}{x^{2}-2x}$$

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

$$a=\frac{2}{x\left(x-2\right)}$$

Solve for x (complex solution)

$$x=\frac{\sqrt{a\left(a+2\right)}+a}{a}$$

$$x=\frac{-\sqrt{a\left(a+2\right)}+a}{a}\text{, }a\neq 0$$

Solve for x

$$x=\frac{\sqrt{a\left(a+2\right)}+a}{a}$$

$$x=\frac{-\sqrt{a\left(a+2\right)}+a}{a}\text{, }a>0\text{ or }a\leq -2$$