Solve 1/x-1/x-2=3

$$\frac{ 1 }{ x } - \frac{ 1 }{ x-2 } =3$$

Solve for x

$$x=\frac{\sqrt{3}}{3}+1\approx 1.577350269$$

$$x=-\frac{\sqrt{3}}{3}+1\approx 0.422649731$$

Steps Using the Quadratic Formula

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), the least common multiple of x,x-2.

$$x-2-x=3x\left(x-2\right)$$

Use the distributive property to multiply 3x by x-2.

$$x-2-x=3x^{2}-6x$$

Subtract 3x^{2} from both sides.

$$x-2-x-3x^{2}=-6x$$

Add 6x to both sides.

$$x-2-x-3x^{2}+6x=0$$

Combine x and 6x to get 7x.

$$7x-2-x-3x^{2}=0$$

Combine 7x and -x to get 6x.

$$6x-2-3x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-3x^{2}+6x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 6 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-3\right)\left(-2\right)}}{2\left(-3\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-3\right)\left(-2\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-6±\sqrt{36+12\left(-2\right)}}{2\left(-3\right)}$$

Multiply 12 times -2.

$$x=\frac{-6±\sqrt{36-24}}{2\left(-3\right)}$$

Add 36 to -24.

$$x=\frac{-6±\sqrt{12}}{2\left(-3\right)}$$

Take the square root of 12.

$$x=\frac{-6±2\sqrt{3}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-6±2\sqrt{3}}{-6}$$

Now solve the equation x=\frac{-6±2\sqrt{3}}{-6} when ± is plus. Add -6 to 2\sqrt{3}.

$$x=\frac{2\sqrt{3}-6}{-6}$$

Divide -6+2\sqrt{3} by -6.

$$x=-\frac{\sqrt{3}}{3}+1$$

Now solve the equation x=\frac{-6±2\sqrt{3}}{-6} when ± is minus. Subtract 2\sqrt{3} from -6.

$$x=\frac{-2\sqrt{3}-6}{-6}$$

Divide -6-2\sqrt{3} by -6.

$$x=\frac{\sqrt{3}}{3}+1$$

The equation is now solved.

$$x=-\frac{\sqrt{3}}{3}+1 x=\frac{\sqrt{3}}{3}+1$$

Steps for Completing the Square

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), the least common multiple of x,x-2.

$$x-2-x=3x\left(x-2\right)$$

Use the distributive property to multiply 3x by x-2.

$$x-2-x=3x^{2}-6x$$

Subtract 3x^{2} from both sides.

$$x-2-x-3x^{2}=-6x$$

Add 6x to both sides.

$$x-2-x-3x^{2}+6x=0$$

Combine x and 6x to get 7x.

$$7x-2-x-3x^{2}=0$$

Add 2 to both sides. Anything plus zero gives itself.

$$7x-x-3x^{2}=2$$

Combine 7x and -x to get 6x.

$$6x-3x^{2}=2$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-3x^{2}+6x=2$$

Divide both sides by -3.

$$\frac{-3x^{2}+6x}{-3}=\frac{2}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{6}{-3}x=\frac{2}{-3}$$

Divide 6 by -3.

$$x^{2}-2x=\frac{2}{-3}$$

Divide 2 by -3.

$$x^{2}-2x=-\frac{2}{3}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=-\frac{2}{3}+1$$

Add -\frac{2}{3} to 1.

$$x^{2}-2x+1=\frac{1}{3}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=\frac{1}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{1}{3}}$$

Simplify.

$$x-1=\frac{\sqrt{3}}{3} x-1=-\frac{\sqrt{3}}{3}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{3}}{3}+1 x=-\frac{\sqrt{3}}{3}+1$$