Solve x+2+x/x-2=3/6

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

Solve for x

$$x=-2$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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 6\left(x-2\right), the least common multiple of x-2,6.

$$6\left(x-2\right)x+6\left(x-2\right)\times 2+6x=\left(x-2\right)\times 3$$

Use the distributive property to multiply 6 by x-2.

$$\left(6x-12\right)x+6\left(x-2\right)\times 2+6x=\left(x-2\right)\times 3$$

Use the distributive property to multiply 6x-12 by x.

$$6x^{2}-12x+6\left(x-2\right)\times 2+6x=\left(x-2\right)\times 3$$

Multiply 6 and 2 to get 12.

$$6x^{2}-12x+12\left(x-2\right)+6x=\left(x-2\right)\times 3$$

Use the distributive property to multiply 12 by x-2.

$$6x^{2}-12x+12x-24+6x=\left(x-2\right)\times 3$$

Combine -12x and 12x to get 0.

$$6x^{2}-24+6x=\left(x-2\right)\times 3$$

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

$$6x^{2}-24+6x=3x-6$$

Subtract 3x from both sides.

$$6x^{2}-24+6x-3x=-6$$

Combine 6x and -3x to get 3x.

$$6x^{2}-24+3x=-6$$

Add 6 to both sides.

$$6x^{2}-24+3x+6=0$$

Add -24 and 6 to get -18.

$$6x^{2}-18+3x=0$$

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.

$$6x^{2}+3x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 3 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 6\left(-18\right)}}{2\times 6}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 6\left(-18\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-3±\sqrt{9-24\left(-18\right)}}{2\times 6}$$

Multiply -24 times -18.

$$x=\frac{-3±\sqrt{9+432}}{2\times 6}$$

Add 9 to 432.

$$x=\frac{-3±\sqrt{441}}{2\times 6}$$

Take the square root of 441.

$$x=\frac{-3±21}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-3±21}{12}$$

Now solve the equation x=\frac{-3±21}{12} when ± is plus. Add -3 to 21.

$$x=\frac{18}{12}$$

Reduce the fraction \frac{18}{12} to lowest terms by extracting and canceling out 6.

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

Now solve the equation x=\frac{-3±21}{12} when ± is minus. Subtract 21 from -3.

$$x=-\frac{24}{12}$$

Divide -24 by 12.

$$x=-2$$

The equation is now solved.

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

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 6\left(x-2\right), the least common multiple of x-2,6.

$$6\left(x-2\right)x+6\left(x-2\right)\times 2+6x=\left(x-2\right)\times 3$$

Use the distributive property to multiply 6 by x-2.

$$\left(6x-12\right)x+6\left(x-2\right)\times 2+6x=\left(x-2\right)\times 3$$

Use the distributive property to multiply 6x-12 by x.

$$6x^{2}-12x+6\left(x-2\right)\times 2+6x=\left(x-2\right)\times 3$$

Multiply 6 and 2 to get 12.

$$6x^{2}-12x+12\left(x-2\right)+6x=\left(x-2\right)\times 3$$

Use the distributive property to multiply 12 by x-2.

$$6x^{2}-12x+12x-24+6x=\left(x-2\right)\times 3$$

Combine -12x and 12x to get 0.

$$6x^{2}-24+6x=\left(x-2\right)\times 3$$

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

$$6x^{2}-24+6x=3x-6$$

Subtract 3x from both sides.

$$6x^{2}-24+6x-3x=-6$$

Combine 6x and -3x to get 3x.

$$6x^{2}-24+3x=-6$$

Add 24 to both sides.

$$6x^{2}+3x=-6+24$$

Add -6 and 24 to get 18.

$$6x^{2}+3x=18$$

Divide both sides by 6.

$$\frac{6x^{2}+3x}{6}=\frac{18}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{3}{6}x=\frac{18}{6}$$

Reduce the fraction \frac{3}{6} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{1}{2}x=\frac{18}{6}$$

Divide 18 by 6.

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

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.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=3+\left(\frac{1}{4}\right)^{2}$$

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

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=3+\frac{1}{16}$$

Add 3 to \frac{1}{16}.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{49}{16}$$

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}.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x+\frac{1}{4}=\frac{7}{4} x+\frac{1}{4}=-\frac{7}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

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