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

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

Solve for x

$$x=\sqrt{6}\approx 2.449489743$$

$$x=-\sqrt{6}\approx -2.449489743$$

Steps by Finding Square Root

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-2,x.

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

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

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

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

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

Subtract x\times 3 from both sides.

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

Combine 3x and -x\times 3 to get 0.

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

Add 6 to both sides. Anything plus zero gives itself.

$$x^{2}=6$$

Take the square root of both sides of the equation.

$$x=\sqrt{6} x=-\sqrt{6}$$

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-2,x.

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

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

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

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

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

Subtract x\times 3 from both sides.

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

Combine 3x and -x\times 3 to get 0.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-6\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-6\right)}}{2}$$

Multiply -4 times -6.

$$x=\frac{0±\sqrt{24}}{2}$$

Take the square root of 24.

$$x=\frac{0±2\sqrt{6}}{2}$$

Now solve the equation x=\frac{0±2\sqrt{6}}{2} when ± is plus.

$$x=\sqrt{6}$$

Now solve the equation x=\frac{0±2\sqrt{6}}{2} when ± is minus.

$$x=-\sqrt{6}$$

The equation is now solved.

$$x=\sqrt{6} x=-\sqrt{6}$$