Solve 3x/2x-2+x/1-x-9/2x+2=0

$$\frac{ 3x }{ 2x-2 } + \frac{ x }{ 1-x } - \frac{ 9 }{ 2x+2 } =0$$

Solve for x

$$x=\sqrt{7}+4\approx 6.645751311$$

$$x=4-\sqrt{7}\approx 1.354248689$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right)\left(x+1\right), the least common multiple of 2x-2,1-x,2x+2.

$$\left(x+1\right)\times 3x+\left(-2-2x\right)x-\left(x-1\right)\times 9=0$$

Use the distributive property to multiply x+1 by 3.

$$\left(3x+3\right)x+\left(-2-2x\right)x-\left(x-1\right)\times 9=0$$

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

$$3x^{2}+3x+\left(-2-2x\right)x-\left(x-1\right)\times 9=0$$

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

$$3x^{2}+3x-2x-2x^{2}-\left(x-1\right)\times 9=0$$

Combine 3x and -2x to get x.

$$3x^{2}+x-2x^{2}-\left(x-1\right)\times 9=0$$

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

$$x^{2}+x-\left(x-1\right)\times 9=0$$

Use the distributive property to multiply x-1 by 9.

$$x^{2}+x-\left(9x-9\right)=0$$

To find the opposite of 9x-9, find the opposite of each term.

$$x^{2}+x-9x+9=0$$

Combine x and -9x to get -8x.

$$x^{2}-8x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 9}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 9}}{2}$$

Multiply -4 times 9.

$$x=\frac{-\left(-8\right)±\sqrt{64-36}}{2}$$

Add 64 to -36.

$$x=\frac{-\left(-8\right)±\sqrt{28}}{2}$$

Take the square root of 28.

$$x=\frac{-\left(-8\right)±2\sqrt{7}}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{7}}{2}$$

Now solve the equation x=\frac{8±2\sqrt{7}}{2} when ± is plus. Add 8 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}+8}{2}$$

Divide 8+2\sqrt{7} by 2.

$$x=\sqrt{7}+4$$

Now solve the equation x=\frac{8±2\sqrt{7}}{2} when ± is minus. Subtract 2\sqrt{7} from 8.

$$x=\frac{8-2\sqrt{7}}{2}$$

Divide 8-2\sqrt{7} by 2.

$$x=4-\sqrt{7}$$

The equation is now solved.

$$x=\sqrt{7}+4 x=4-\sqrt{7}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-1\right)\left(x+1\right), the least common multiple of 2x-2,1-x,2x+2.

$$\left(x+1\right)\times 3x+\left(-2-2x\right)x-\left(x-1\right)\times 9=0$$

Use the distributive property to multiply x+1 by 3.

$$\left(3x+3\right)x+\left(-2-2x\right)x-\left(x-1\right)\times 9=0$$

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

$$3x^{2}+3x+\left(-2-2x\right)x-\left(x-1\right)\times 9=0$$

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

$$3x^{2}+3x-2x-2x^{2}-\left(x-1\right)\times 9=0$$

Combine 3x and -2x to get x.

$$3x^{2}+x-2x^{2}-\left(x-1\right)\times 9=0$$

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

$$x^{2}+x-\left(x-1\right)\times 9=0$$

Use the distributive property to multiply x-1 by 9.

$$x^{2}+x-\left(9x-9\right)=0$$

To find the opposite of 9x-9, find the opposite of each term.

$$x^{2}+x-9x+9=0$$

Combine x and -9x to get -8x.

$$x^{2}-8x+9=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-8x=-9$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-9+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-9+16$$

Add -9 to 16.

$$x^{2}-8x+16=7$$

Factor x^{2}-8x+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}.

$$\left(x-4\right)^{2}=7$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{7}$$

Simplify.

$$x-4=\sqrt{7} x-4=-\sqrt{7}$$

Add 4 to both sides of the equation.

$$x=\sqrt{7}+4 x=4-\sqrt{7}$$