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

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

Solve for x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x=0$$

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

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

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

$$\left(2x+4\right)\times 2-2\left(x-2\right)=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)$$

Use the distributive property to multiply 2x+4 by 2.

$$4x+8-2\left(x-2\right)=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)$$

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

$$4x+8-2x+4=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)$$

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

$$2x+8+4=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)$$

Add 8 and 4 to get 12.

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

Multiply \frac{3}{2} and 2 to get 3.

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

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

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

Use the distributive property to multiply 3\left(-x\right)+6 by x+2.

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

Subtract 3\left(-x\right)x from both sides.

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

Subtract 6\left(-x\right) from both sides.

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

Subtract 6x from both sides.

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

Subtract 12 from both sides.

$$2x+12-3\left(-x\right)x-6\left(-x\right)-6x-12=0$$

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

$$2x+12-3\left(-1\right)x^{2}-6\left(-1\right)x-6x-12=0$$

Multiply -3 and -1 to get 3.

$$2x+12+3x^{2}-6\left(-1\right)x-6x-12=0$$

Multiply -6 and -1 to get 6.

$$2x+12+3x^{2}+6x-6x-12=0$$

Combine 2x and 6x to get 8x.

$$8x+12+3x^{2}-6x-12=0$$

Combine 8x and -6x to get 2x.

$$2x+12+3x^{2}-12=0$$

Subtract 12 from 12 to get 0.

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

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.

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

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

$$x=\frac{-2±\sqrt{2^{2}}}{2\times 3}$$

Take the square root of 2^{2}.

$$x=\frac{-2±2}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-2±2}{6}$$

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

$$x=\frac{0}{6}$$

Divide 0 by 6.

$$x=0$$

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

$$x=-\frac{4}{6}$$

Reduce the fraction \frac{-4}{6} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

$$x=0 x=-\frac{2}{3}$$

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

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

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

$$\left(2x+4\right)\times 2-2\left(x-2\right)=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)$$

Use the distributive property to multiply 2x+4 by 2.

$$4x+8-2\left(x-2\right)=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)$$

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

$$4x+8-2x+4=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)$$

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

$$2x+8+4=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)$$

Add 8 and 4 to get 12.

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

Multiply \frac{3}{2} and 2 to get 3.

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

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

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

Use the distributive property to multiply 3\left(-x\right)+6 by x+2.

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

Subtract 3\left(-x\right)x from both sides.

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

Subtract 6\left(-x\right) from both sides.

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

Subtract 6x from both sides.

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

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

$$2x+12-3\left(-1\right)x^{2}-6\left(-1\right)x-6x=12$$

Multiply -3 and -1 to get 3.

$$2x+12+3x^{2}-6\left(-1\right)x-6x=12$$

Multiply -6 and -1 to get 6.

$$2x+12+3x^{2}+6x-6x=12$$

Combine 2x and 6x to get 8x.

$$8x+12+3x^{2}-6x=12$$

Combine 8x and -6x to get 2x.

$$2x+12+3x^{2}=12$$

Subtract 12 from both sides.

$$2x+3x^{2}=12-12$$

Subtract 12 from 12 to get 0.

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

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.

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

Divide both sides by 3.

$$\frac{3x^{2}+2x}{3}=\frac{0}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{2}{3}x=\frac{0}{3}$$

Divide 0 by 3.

$$x^{2}+\frac{2}{3}x=0$$

Divide \frac{2}{3}, the coefficient of the x term, by 2 to get \frac{1}{3}. Then add the square of \frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=\frac{1}{9}$$

Factor x^{2}+\frac{2}{3}x+\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{3}\right)^{2}=\frac{1}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{3}\right)^{2}}=\sqrt{\frac{1}{9}}$$

Simplify.

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

Subtract \frac{1}{3} from both sides of the equation.

$$x=0 x=-\frac{2}{3}$$