Solve x+1/x+x/x+1=27/6

$$\frac { x + 1 } { x } + \frac { x } { x + 1 } = 2 \frac { 7 } { 6 }$$

Solve for x

$$x=-\frac{\sqrt{217}}{14}-\frac{1}{2}\approx -1.552208562$$

$$x=\frac{\sqrt{217}}{14}-\frac{1}{2}\approx 0.552208562$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+1\right), the least common multiple of x,x+1,6.

$$\left(6x+6\right)\left(x+1\right)+6xx=x\left(x+1\right)\left(2\times 6+7\right)$$

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

$$6x^{2}+12x+6+6xx=x\left(x+1\right)\left(2\times 6+7\right)$$

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

$$6x^{2}+12x+6+6x^{2}=x\left(x+1\right)\left(2\times 6+7\right)$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$12x^{2}+12x+6=x\left(x+1\right)\left(2\times 6+7\right)$$

Multiply 2 and 6 to get 12.

$$12x^{2}+12x+6=x\left(x+1\right)\left(12+7\right)$$

Add 12 and 7 to get 19.

$$12x^{2}+12x+6=x\left(x+1\right)\times 19$$

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

$$12x^{2}+12x+6=\left(x^{2}+x\right)\times 19$$

Use the distributive property to multiply x^{2}+x by 19.

$$12x^{2}+12x+6=19x^{2}+19x$$

Subtract 19x^{2} from both sides.

$$12x^{2}+12x+6-19x^{2}=19x$$

Combine 12x^{2} and -19x^{2} to get -7x^{2}.

$$-7x^{2}+12x+6=19x$$

Subtract 19x from both sides.

$$-7x^{2}+12x+6-19x=0$$

Combine 12x and -19x to get -7x.

$$-7x^{2}-7x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, -7 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-7\right)\times 6}}{2\left(-7\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-7\right)\times 6}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-\left(-7\right)±\sqrt{49+28\times 6}}{2\left(-7\right)}$$

Multiply 28 times 6.

$$x=\frac{-\left(-7\right)±\sqrt{49+168}}{2\left(-7\right)}$$

Add 49 to 168.

$$x=\frac{-\left(-7\right)±\sqrt{217}}{2\left(-7\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{217}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{7±\sqrt{217}}{-14}$$

Now solve the equation x=\frac{7±\sqrt{217}}{-14} when ± is plus. Add 7 to \sqrt{217}.

$$x=\frac{\sqrt{217}+7}{-14}$$

Divide 7+\sqrt{217} by -14.

$$x=-\frac{\sqrt{217}}{14}-\frac{1}{2}$$

Now solve the equation x=\frac{7±\sqrt{217}}{-14} when ± is minus. Subtract \sqrt{217} from 7.

$$x=\frac{7-\sqrt{217}}{-14}$$

Divide 7-\sqrt{217} by -14.

$$x=\frac{\sqrt{217}}{14}-\frac{1}{2}$$

The equation is now solved.

$$x=-\frac{\sqrt{217}}{14}-\frac{1}{2} x=\frac{\sqrt{217}}{14}-\frac{1}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+1\right), the least common multiple of x,x+1,6.

$$\left(6x+6\right)\left(x+1\right)+6xx=x\left(x+1\right)\left(2\times 6+7\right)$$

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

$$6x^{2}+12x+6+6xx=x\left(x+1\right)\left(2\times 6+7\right)$$

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

$$6x^{2}+12x+6+6x^{2}=x\left(x+1\right)\left(2\times 6+7\right)$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$12x^{2}+12x+6=x\left(x+1\right)\left(2\times 6+7\right)$$

Multiply 2 and 6 to get 12.

$$12x^{2}+12x+6=x\left(x+1\right)\left(12+7\right)$$

Add 12 and 7 to get 19.

$$12x^{2}+12x+6=x\left(x+1\right)\times 19$$

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

$$12x^{2}+12x+6=\left(x^{2}+x\right)\times 19$$

Use the distributive property to multiply x^{2}+x by 19.

$$12x^{2}+12x+6=19x^{2}+19x$$

Subtract 19x^{2} from both sides.

$$12x^{2}+12x+6-19x^{2}=19x$$

Combine 12x^{2} and -19x^{2} to get -7x^{2}.

$$-7x^{2}+12x+6=19x$$

Subtract 19x from both sides.

$$-7x^{2}+12x+6-19x=0$$

Combine 12x and -19x to get -7x.

$$-7x^{2}-7x+6=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$-7x^{2}-7x=-6$$

Divide both sides by -7.

$$\frac{-7x^{2}-7x}{-7}=-\frac{6}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\left(-\frac{7}{-7}\right)x=-\frac{6}{-7}$$

Divide -7 by -7.

$$x^{2}+x=-\frac{6}{-7}$$

Divide -6 by -7.

$$x^{2}+x=\frac{6}{7}$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{6}{7}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=\frac{6}{7}+\frac{1}{4}$$

Add \frac{6}{7} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{31}{28}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{31}{28}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{31}{28}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{217}}{14} x+\frac{1}{2}=-\frac{\sqrt{217}}{14}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{217}}{14}-\frac{1}{2} x=-\frac{\sqrt{217}}{14}-\frac{1}{2}$$