Solve x+1/2-2x-2x-1/x^2-1+6/x+1+1/2-2x=0

$$\frac{ x+1 }{ 2-2x } - \frac{ 2x-1 }{ { x }^{ 2 } -1 } + \frac{ 6 }{ x+1 } + \frac{ 1 }{ 2-2x } =0$$

Solve for x (complex solution)

$$x=\frac{5+\sqrt{23}i}{2}\approx 2.5+2.397915762i$$

$$x=\frac{-\sqrt{23}i+5}{2}\approx 2.5-2.397915762i$$

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 2-2x,x^{2}-1,x+1.

$$-\left(1+x\right)\left(x+1\right)-2\left(2x-1\right)+\left(2x-2\right)\times 6-\left(1+x\right)=0$$

To find the opposite of 1+x, find the opposite of each term.

$$\left(-1-x\right)\left(x+1\right)-2\left(2x-1\right)+\left(2x-2\right)\times 6-\left(1+x\right)=0$$

Use the distributive property to multiply -1-x by x+1 and combine like terms.

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

Use the distributive property to multiply -2 by 2x-1.

$$-2x-1-x^{2}-4x+2+\left(2x-2\right)\times 6-\left(1+x\right)=0$$

Combine -2x and -4x to get -6x.

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

Add -1 and 2 to get 1.

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

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

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

Combine -6x and 12x to get 6x.

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

Subtract 12 from 1 to get -11.

$$6x-11-x^{2}-\left(1+x\right)=0$$

To find the opposite of 1+x, find the opposite of each term.

$$6x-11-x^{2}-1-x=0$$

Subtract 1 from -11 to get -12.

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

Combine 6x and -x to get 5x.

$$5x-12-x^{2}=0$$

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.

$$-x^{2}+5x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 5 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\left(-12\right)}}{2\left(-1\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\left(-12\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-5±\sqrt{25+4\left(-12\right)}}{2\left(-1\right)}$$

Multiply 4 times -12.

$$x=\frac{-5±\sqrt{25-48}}{2\left(-1\right)}$$

Add 25 to -48.

$$x=\frac{-5±\sqrt{-23}}{2\left(-1\right)}$$

Take the square root of -23.

$$x=\frac{-5±\sqrt{23}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-5±\sqrt{23}i}{-2}$$

Now solve the equation x=\frac{-5±\sqrt{23}i}{-2} when ± is plus. Add -5 to i\sqrt{23}.

$$x=\frac{-5+\sqrt{23}i}{-2}$$

Divide -5+i\sqrt{23} by -2.

$$x=\frac{-\sqrt{23}i+5}{2}$$

Now solve the equation x=\frac{-5±\sqrt{23}i}{-2} when ± is minus. Subtract i\sqrt{23} from -5.

$$x=\frac{-\sqrt{23}i-5}{-2}$$

Divide -5-i\sqrt{23} by -2.

$$x=\frac{5+\sqrt{23}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{23}i+5}{2} x=\frac{5+\sqrt{23}i}{2}$$

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 2-2x,x^{2}-1,x+1.

$$-\left(1+x\right)\left(x+1\right)-2\left(2x-1\right)+\left(2x-2\right)\times 6-\left(1+x\right)=0$$

To find the opposite of 1+x, find the opposite of each term.

$$\left(-1-x\right)\left(x+1\right)-2\left(2x-1\right)+\left(2x-2\right)\times 6-\left(1+x\right)=0$$

Use the distributive property to multiply -1-x by x+1 and combine like terms.

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

Use the distributive property to multiply -2 by 2x-1.

$$-2x-1-x^{2}-4x+2+\left(2x-2\right)\times 6-\left(1+x\right)=0$$

Combine -2x and -4x to get -6x.

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

Add -1 and 2 to get 1.

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

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

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

Combine -6x and 12x to get 6x.

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

Subtract 12 from 1 to get -11.

$$6x-11-x^{2}-\left(1+x\right)=0$$

To find the opposite of 1+x, find the opposite of each term.

$$6x-11-x^{2}-1-x=0$$

Subtract 1 from -11 to get -12.

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

Combine 6x and -x to get 5x.

$$5x-12-x^{2}=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$5x-x^{2}=12$$

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.

$$-x^{2}+5x=12$$

Divide both sides by -1.

$$\frac{-x^{2}+5x}{-1}=\frac{12}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{5}{-1}x=\frac{12}{-1}$$

Divide 5 by -1.

$$x^{2}-5x=\frac{12}{-1}$$

Divide 12 by -1.

$$x^{2}-5x=-12$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-12+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-12+\frac{25}{4}$$

Add -12 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=-\frac{23}{4}$$

Factor x^{2}-5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{2}\right)^{2}=-\frac{23}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{23}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{23}i}{2} x-\frac{5}{2}=-\frac{\sqrt{23}i}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{5+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+5}{2}$$