Solve 2/1-x+12/(x-1)(x+1)-1-3/x-1=0

$$\frac { 2 } { 1 - x } + \frac { 12 } { ( x - 1 ) ( x + 1 ) } - 1 - \frac { 3 } { x - 1 } = 0$$

Solve for x

$$x = \frac{\sqrt{57} - 5}{2} \approx 1.274917218$$

$$x=\frac{-\sqrt{57}-5}{2}\approx -6.274917218$$

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 \left(x-1\right)\left(x+1\right), the least common multiple of 1-x,\left(x-1\right)\left(x+1\right),x-1.

$$-\left(1+x\right)\times 2+12+\left(x-1\right)\left(x+1\right)\left(-1\right)-\left(x+1\right)\times 3=0$$

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

$$\left(-1-x\right)\times 2+12+\left(x-1\right)\left(x+1\right)\left(-1\right)-\left(x+1\right)\times 3=0$$

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

$$-2-2x+12+\left(x-1\right)\left(x+1\right)\left(-1\right)-\left(x+1\right)\times 3=0$$

Add -2 and 12 to get 10.

$$10-2x+\left(x-1\right)\left(x+1\right)\left(-1\right)-\left(x+1\right)\times 3=0$$

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

$$10-2x+\left(x^{2}-1\right)\left(-1\right)-\left(x+1\right)\times 3=0$$

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

$$10-2x-x^{2}+1-\left(x+1\right)\times 3=0$$

Add 10 and 1 to get 11.

$$11-2x-x^{2}-\left(x+1\right)\times 3=0$$

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

$$11-2x-x^{2}-\left(3x+3\right)=0$$

To find the opposite of 3x+3, find the opposite of each term.

$$11-2x-x^{2}-3x-3=0$$

Combine -2x and -3x to get -5x.

$$11-5x-x^{2}-3=0$$

Subtract 3 from 11 to get 8.

$$8-5x-x^{2}=0$$

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.

$$-x^{2}-5x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -5 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-5\right)±\sqrt{25+4\times 8}}{2\left(-1\right)}$$

Multiply 4 times 8.

$$x=\frac{-\left(-5\right)±\sqrt{25+32}}{2\left(-1\right)}$$

Add 25 to 32.

$$x=\frac{-\left(-5\right)±\sqrt{57}}{2\left(-1\right)}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{57}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{5±\sqrt{57}}{-2}$$

Now solve the equation x=\frac{5±\sqrt{57}}{-2} when ± is plus. Add 5 to \sqrt{57}.

$$x=\frac{\sqrt{57}+5}{-2}$$

Divide 5+\sqrt{57} by -2.

$$x=\frac{-\sqrt{57}-5}{2}$$

Now solve the equation x=\frac{5±\sqrt{57}}{-2} when ± is minus. Subtract \sqrt{57} from 5.

$$x=\frac{5-\sqrt{57}}{-2}$$

Divide 5-\sqrt{57} by -2.

$$x=\frac{\sqrt{57}-5}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{57}-5}{2} x=\frac{\sqrt{57}-5}{2}$$

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 \left(x-1\right)\left(x+1\right), the least common multiple of 1-x,\left(x-1\right)\left(x+1\right),x-1.

$$-\left(1+x\right)\times 2+12+\left(x-1\right)\left(x+1\right)\left(-1\right)-\left(x+1\right)\times 3=0$$

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

$$\left(-1-x\right)\times 2+12+\left(x-1\right)\left(x+1\right)\left(-1\right)-\left(x+1\right)\times 3=0$$

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

$$-2-2x+12+\left(x-1\right)\left(x+1\right)\left(-1\right)-\left(x+1\right)\times 3=0$$

Add -2 and 12 to get 10.

$$10-2x+\left(x-1\right)\left(x+1\right)\left(-1\right)-\left(x+1\right)\times 3=0$$

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

$$10-2x+\left(x^{2}-1\right)\left(-1\right)-\left(x+1\right)\times 3=0$$

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

$$10-2x-x^{2}+1-\left(x+1\right)\times 3=0$$

Add 10 and 1 to get 11.

$$11-2x-x^{2}-\left(x+1\right)\times 3=0$$

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

$$11-2x-x^{2}-\left(3x+3\right)=0$$

To find the opposite of 3x+3, find the opposite of each term.

$$11-2x-x^{2}-3x-3=0$$

Combine -2x and -3x to get -5x.

$$11-5x-x^{2}-3=0$$

Subtract 3 from 11 to get 8.

$$8-5x-x^{2}=0$$

Subtract 8 from both sides. Anything subtracted from zero gives its negation.

$$-5x-x^{2}=-8$$

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.

$$-x^{2}-5x=-8$$

Divide both sides by -1.

$$\frac{-x^{2}-5x}{-1}=-\frac{8}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{5}{-1}\right)x=-\frac{8}{-1}$$

Divide -5 by -1.

$$x^{2}+5x=-\frac{8}{-1}$$

Divide -8 by -1.

$$x^{2}+5x=8$$

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.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=8+\left(\frac{5}{2}\right)^{2}$$

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

$$x^{2}+5x+\frac{25}{4}=8+\frac{25}{4}$$

Add 8 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{57}{4}$$

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}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{57}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{57}{4}}$$

Simplify.

$$x+\frac{5}{2}=\frac{\sqrt{57}}{2} x+\frac{5}{2}=-\frac{\sqrt{57}}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=\frac{\sqrt{57}-5}{2} x=\frac{-\sqrt{57}-5}{2}$$