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

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

Solve for x

$$x=\frac{1}{2}=0.5$$

Steps Using Factoring

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.

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

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

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

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

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

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

$$x^{2}-2x-x^{2}\times 3=2x^{2}-4x$$

Subtract 2x^{2} from both sides.

$$x^{2}-2x-x^{2}\times 3-2x^{2}=-4x$$

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

$$-x^{2}-2x-x^{2}\times 3=-4x$$

Add 4x to both sides.

$$-x^{2}-2x-x^{2}\times 3+4x=0$$

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

$$-x^{2}+2x-x^{2}\times 3=0$$

Multiply -1 and 3 to get -3.

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

Combine -x^{2} and -3x^{2} to get -4x^{2}.

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

Factor out x.

$$x\left(-4x+2\right)=0$$

To find equation solutions, solve x=0 and -4x+2=0.

$$x=0 x=\frac{1}{2}$$

Variable x cannot be equal to 0.

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

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.

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

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

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

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

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

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

$$x^{2}-2x-x^{2}\times 3=2x^{2}-4x$$

Subtract 2x^{2} from both sides.

$$x^{2}-2x-x^{2}\times 3-2x^{2}=-4x$$

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

$$-x^{2}-2x-x^{2}\times 3=-4x$$

Add 4x to both sides.

$$-x^{2}-2x-x^{2}\times 3+4x=0$$

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

$$-x^{2}+2x-x^{2}\times 3=0$$

Multiply -1 and 3 to get -3.

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

Combine -x^{2} and -3x^{2} to get -4x^{2}.

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

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

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

Take the square root of 2^{2}.

$$x=\frac{-2±2}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-2±2}{-8}$$

Now solve the equation x=\frac{-2±2}{-8} when ± is plus. Add -2 to 2.

$$x=\frac{0}{-8}$$

Divide 0 by -8.

$$x=0$$

Now solve the equation x=\frac{-2±2}{-8} when ± is minus. Subtract 2 from -2.

$$x=-\frac{4}{-8}$$

Reduce the fraction \frac{-4}{-8} to lowest terms by extracting and canceling out 4.

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

The equation is now solved.

$$x=0 x=\frac{1}{2}$$

Variable x cannot be equal to 0.

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

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.

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

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

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

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

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

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

$$x^{2}-2x-x^{2}\times 3=2x^{2}-4x$$

Subtract 2x^{2} from both sides.

$$x^{2}-2x-x^{2}\times 3-2x^{2}=-4x$$

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

$$-x^{2}-2x-x^{2}\times 3=-4x$$

Add 4x to both sides.

$$-x^{2}-2x-x^{2}\times 3+4x=0$$

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

$$-x^{2}+2x-x^{2}\times 3=0$$

Multiply -1 and 3 to get -3.

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

Combine -x^{2} and -3x^{2} to get -4x^{2}.

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

Divide both sides by -4.

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

Dividing by -4 undoes the multiplication by -4.

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

Reduce the fraction \frac{2}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{1}{2}x=\frac{0}{-4}$$

Divide 0 by -4.

$$x^{2}-\frac{1}{2}x=0$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{1}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{1}{4} x-\frac{1}{4}=-\frac{1}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{1}{2} x=0$$

Variable x cannot be equal to 0.

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