Solve x+1/x+x/x+1=5/2

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

Solve for x

$$x=1$$

$$x=-2$$

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 2x\left(x+1\right), the least common multiple of x,x+1,2.

$$\left(2x+2\right)\left(x+1\right)+2xx=5x\left(x+1\right)$$

Use the distributive property to multiply 2x+2 by x+1 and combine like terms.

$$2x^{2}+4x+2+2xx=5x\left(x+1\right)$$

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

$$2x^{2}+4x+2+2x^{2}=5x\left(x+1\right)$$

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$4x^{2}+4x+2=5x\left(x+1\right)$$

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

$$4x^{2}+4x+2=5x^{2}+5x$$

Subtract 5x^{2} from both sides.

$$4x^{2}+4x+2-5x^{2}=5x$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}+4x+2=5x$$

Subtract 5x from both sides.

$$-x^{2}+4x+2-5x=0$$

Combine 4x and -5x to get -x.

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

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

$$a+b=-1 ab=-2=-2$$

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. The only such pair is the system solution.

$$a=1 b=-2$$

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

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

Factor out x in the first and 2 in the second group.

$$x\left(-x+1\right)+2\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(x+2\right)$$

To find equation solutions, solve -x+1=0 and x+2=0.

$$x=1 x=-2$$

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 2x\left(x+1\right), the least common multiple of x,x+1,2.

$$\left(2x+2\right)\left(x+1\right)+2xx=5x\left(x+1\right)$$

Use the distributive property to multiply 2x+2 by x+1 and combine like terms.

$$2x^{2}+4x+2+2xx=5x\left(x+1\right)$$

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

$$2x^{2}+4x+2+2x^{2}=5x\left(x+1\right)$$

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$4x^{2}+4x+2=5x\left(x+1\right)$$

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

$$4x^{2}+4x+2=5x^{2}+5x$$

Subtract 5x^{2} from both sides.

$$4x^{2}+4x+2-5x^{2}=5x$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}+4x+2=5x$$

Subtract 5x from both sides.

$$-x^{2}+4x+2-5x=0$$

Combine 4x and -5x to get -x.

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

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

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

Multiply -4 times -1.

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

Multiply 4 times 2.

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

Add 1 to 8.

$$x=\frac{-\left(-1\right)±\sqrt{9}}{2\left(-1\right)}$$

Take the square root of 9.

$$x=\frac{-\left(-1\right)±3}{2\left(-1\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±3}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{1±3}{-2}$$

Now solve the equation x=\frac{1±3}{-2} when ± is plus. Add 1 to 3.

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

Divide 4 by -2.

$$x=-2$$

Now solve the equation x=\frac{1±3}{-2} when ± is minus. Subtract 3 from 1.

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

Divide -2 by -2.

$$x=1$$

The equation is now solved.

$$x=-2 x=1$$

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 2x\left(x+1\right), the least common multiple of x,x+1,2.

$$\left(2x+2\right)\left(x+1\right)+2xx=5x\left(x+1\right)$$

Use the distributive property to multiply 2x+2 by x+1 and combine like terms.

$$2x^{2}+4x+2+2xx=5x\left(x+1\right)$$

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

$$2x^{2}+4x+2+2x^{2}=5x\left(x+1\right)$$

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$4x^{2}+4x+2=5x\left(x+1\right)$$

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

$$4x^{2}+4x+2=5x^{2}+5x$$

Subtract 5x^{2} from both sides.

$$4x^{2}+4x+2-5x^{2}=5x$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}+4x+2=5x$$

Subtract 5x from both sides.

$$-x^{2}+4x+2-5x=0$$

Combine 4x and -5x to get -x.

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

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide -1 by -1.

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

Divide -2 by -1.

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

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.

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

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

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

Add 2 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{9}{4}$$

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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{3}{2} x+\frac{1}{2}=-\frac{3}{2}$$

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

$$x=1 x=-2$$