Solve x+1/x-1-x-1/x+1=5/6

$$\frac { x + 1 } { x - 1 } - \frac { x - 1 } { x + 1 } = \frac { 5 } { 6 }$$

Solve for x

$$x=5$$

$$x=-\frac{1}{5}=-0.2$$

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 6\left(x-1\right)\left(x+1\right), the least common multiple of x-1,x+1,6.

$$\left(6x+6\right)\left(x+1\right)-\left(6x-6\right)\left(x-1\right)=5\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 6x+6 by x+1 and combine like terms.

$$6x^{2}+12x+6-\left(6x-6\right)\left(x-1\right)=5\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 6x-6 by x-1 and combine like terms.

$$6x^{2}+12x+6-\left(6x^{2}-12x+6\right)=5\left(x-1\right)\left(x+1\right)$$

To find the opposite of 6x^{2}-12x+6, find the opposite of each term.

$$6x^{2}+12x+6-6x^{2}+12x-6=5\left(x-1\right)\left(x+1\right)$$

Combine 6x^{2} and -6x^{2} to get 0.

$$12x+6+12x-6=5\left(x-1\right)\left(x+1\right)$$

Combine 12x and 12x to get 24x.

$$24x+6-6=5\left(x-1\right)\left(x+1\right)$$

Subtract 6 from 6 to get 0.

$$24x=5\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 5 by x-1.

$$24x=\left(5x-5\right)\left(x+1\right)$$

Use the distributive property to multiply 5x-5 by x+1 and combine like terms.

$$24x=5x^{2}-5$$

Subtract 5x^{2} from both sides.

$$24x-5x^{2}=-5$$

Add 5 to both sides.

$$24x-5x^{2}+5=0$$

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.

$$-5x^{2}+24x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 24 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-5\right)\times 5}}{2\left(-5\right)}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-5\right)\times 5}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-24±\sqrt{576+20\times 5}}{2\left(-5\right)}$$

Multiply 20 times 5.

$$x=\frac{-24±\sqrt{576+100}}{2\left(-5\right)}$$

Add 576 to 100.

$$x=\frac{-24±\sqrt{676}}{2\left(-5\right)}$$

Take the square root of 676.

$$x=\frac{-24±26}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-24±26}{-10}$$

Now solve the equation x=\frac{-24±26}{-10} when ± is plus. Add -24 to 26.

$$x=\frac{2}{-10}$$

Reduce the fraction \frac{2}{-10} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{5}$$

Now solve the equation x=\frac{-24±26}{-10} when ± is minus. Subtract 26 from -24.

$$x=-\frac{50}{-10}$$

Divide -50 by -10.

$$x=5$$

The equation is now solved.

$$x=-\frac{1}{5} x=5$$

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 6\left(x-1\right)\left(x+1\right), the least common multiple of x-1,x+1,6.

$$\left(6x+6\right)\left(x+1\right)-\left(6x-6\right)\left(x-1\right)=5\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 6x+6 by x+1 and combine like terms.

$$6x^{2}+12x+6-\left(6x-6\right)\left(x-1\right)=5\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 6x-6 by x-1 and combine like terms.

$$6x^{2}+12x+6-\left(6x^{2}-12x+6\right)=5\left(x-1\right)\left(x+1\right)$$

To find the opposite of 6x^{2}-12x+6, find the opposite of each term.

$$6x^{2}+12x+6-6x^{2}+12x-6=5\left(x-1\right)\left(x+1\right)$$

Combine 6x^{2} and -6x^{2} to get 0.

$$12x+6+12x-6=5\left(x-1\right)\left(x+1\right)$$

Combine 12x and 12x to get 24x.

$$24x+6-6=5\left(x-1\right)\left(x+1\right)$$

Subtract 6 from 6 to get 0.

$$24x=5\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 5 by x-1.

$$24x=\left(5x-5\right)\left(x+1\right)$$

Use the distributive property to multiply 5x-5 by x+1 and combine like terms.

$$24x=5x^{2}-5$$

Subtract 5x^{2} from both sides.

$$24x-5x^{2}=-5$$

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.

$$-5x^{2}+24x=-5$$

Divide both sides by -5.

$$\frac{-5x^{2}+24x}{-5}=-\frac{5}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{24}{-5}x=-\frac{5}{-5}$$

Divide 24 by -5.

$$x^{2}-\frac{24}{5}x=-\frac{5}{-5}$$

Divide -5 by -5.

$$x^{2}-\frac{24}{5}x=1$$

Divide -\frac{24}{5}, the coefficient of the x term, by 2 to get -\frac{12}{5}. Then add the square of -\frac{12}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{24}{5}x+\left(-\frac{12}{5}\right)^{2}=1+\left(-\frac{12}{5}\right)^{2}$$

Square -\frac{12}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{24}{5}x+\frac{144}{25}=1+\frac{144}{25}$$

Add 1 to \frac{144}{25}.

$$x^{2}-\frac{24}{5}x+\frac{144}{25}=\frac{169}{25}$$

Factor x^{2}-\frac{24}{5}x+\frac{144}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{12}{5}\right)^{2}=\frac{169}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{12}{5}\right)^{2}}=\sqrt{\frac{169}{25}}$$

Simplify.

$$x-\frac{12}{5}=\frac{13}{5} x-\frac{12}{5}=-\frac{13}{5}$$

Add \frac{12}{5} to both sides of the equation.

$$x=5 x=-\frac{1}{5}$$