Solve x-2=4/x-2

$$x-2 = \frac{ 4 }{ x-2 }$$

Solve for x

$$x=0$$

$$x=4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

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

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

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

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

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

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

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

Subtract 4 from both sides.

$$x^{2}-4x+4-4=0$$

Subtract 4 from 4 to get 0.

$$x^{2}-4x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}}}{2}$$

Take the square root of \left(-4\right)^{2}.

$$x=\frac{-\left(-4\right)±4}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±4}{2}$$

Now solve the equation x=\frac{4±4}{2} when ± is plus. Add 4 to 4.

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

Divide 8 by 2.

$$x=4$$

Now solve the equation x=\frac{4±4}{2} when ± is minus. Subtract 4 from 4.

$$x=\frac{0}{2}$$

Divide 0 by 2.

$$x=0$$

The equation is now solved.

$$x=4 x=0$$

Steps for Completing the Square

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

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

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

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

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

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

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

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

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-2=2 x-2=-2$$

Add 2 to both sides of the equation.

$$x=4 x=0$$