Solve x/x+1+x+1/x=13/6

$$\frac { x } { x + 1 } + \frac { x + 1 } { x } = \frac { 13 } { 6 }$$

Solve for x

$$x=-3$$

$$x=2$$

Steps Using Factoring By Grouping

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+1,x,6.

$$6xx+\left(6x+6\right)\left(x+1\right)=13x\left(x+1\right)$$

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

$$6x^{2}+\left(6x+6\right)\left(x+1\right)=13x\left(x+1\right)$$

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

$$6x^{2}+6x^{2}+12x+6=13x\left(x+1\right)$$

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

$$12x^{2}+12x+6=13x\left(x+1\right)$$

Use the distributive property to multiply 13x by x+1.

$$12x^{2}+12x+6=13x^{2}+13x$$

Subtract 13x^{2} from both sides.

$$12x^{2}+12x+6-13x^{2}=13x$$

Combine 12x^{2} and -13x^{2} to get -x^{2}.

$$-x^{2}+12x+6=13x$$

Subtract 13x from both sides.

$$-x^{2}+12x+6-13x=0$$

Combine 12x and -13x to get -x.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=-1 ab=-6=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -1.

$$a=2 b=-3$$

Rewrite -x^{2}-x+6 as \left(-x^{2}+2x\right)+\left(-3x+6\right).

$$\left(-x^{2}+2x\right)+\left(-3x+6\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(-x+2\right)+3\left(-x+2\right)$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(x+3\right)$$

To find equation solutions, solve -x+2=0 and x+3=0.

$$x=2 x=-3$$

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+1,x,6.

$$6xx+\left(6x+6\right)\left(x+1\right)=13x\left(x+1\right)$$

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

$$6x^{2}+\left(6x+6\right)\left(x+1\right)=13x\left(x+1\right)$$

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

$$6x^{2}+6x^{2}+12x+6=13x\left(x+1\right)$$

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

$$12x^{2}+12x+6=13x\left(x+1\right)$$

Use the distributive property to multiply 13x by x+1.

$$12x^{2}+12x+6=13x^{2}+13x$$

Subtract 13x^{2} from both sides.

$$12x^{2}+12x+6-13x^{2}=13x$$

Combine 12x^{2} and -13x^{2} to get -x^{2}.

$$-x^{2}+12x+6=13x$$

Subtract 13x from both sides.

$$-x^{2}+12x+6-13x=0$$

Combine 12x and -13x to get -x.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -1 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\times 6}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+4\times 6}}{2\left(-1\right)}$$

Multiply 4 times 6.

$$x=\frac{-\left(-1\right)±\sqrt{1+24}}{2\left(-1\right)}$$

Add 1 to 24.

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

Take the square root of 25.

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

The opposite of -1 is 1.

$$x=\frac{1±5}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{1±5}{-2}$$

Now solve the equation x=\frac{1±5}{-2} when ± is plus. Add 1 to 5.

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

Divide 6 by -2.

$$x=-3$$

Now solve the equation x=\frac{1±5}{-2} when ± is minus. Subtract 5 from 1.

$$x=-\frac{4}{-2}$$

Divide -4 by -2.

$$x=2$$

The equation is now solved.

$$x=-3 x=2$$

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+1,x,6.

$$6xx+\left(6x+6\right)\left(x+1\right)=13x\left(x+1\right)$$

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

$$6x^{2}+\left(6x+6\right)\left(x+1\right)=13x\left(x+1\right)$$

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

$$6x^{2}+6x^{2}+12x+6=13x\left(x+1\right)$$

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

$$12x^{2}+12x+6=13x\left(x+1\right)$$

Use the distributive property to multiply 13x by x+1.

$$12x^{2}+12x+6=13x^{2}+13x$$

Subtract 13x^{2} from both sides.

$$12x^{2}+12x+6-13x^{2}=13x$$

Combine 12x^{2} and -13x^{2} to get -x^{2}.

$$-x^{2}+12x+6=13x$$

Subtract 13x from both sides.

$$-x^{2}+12x+6-13x=0$$

Combine 12x and -13x to get -x.

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

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

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide -1 by -1.

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

Divide -6 by -1.

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

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.

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

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

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

Add 6 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{25}{4}$$

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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

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

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

$$x=2 x=-3$$