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

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

Solve for x

$$x=-4$$

$$x=2$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+2\right), the least common multiple of x+2,x,2.

$$2xx+\left(2x+4\right)\left(x+2\right)=5x\left(x+2\right)$$

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

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

Use the distributive property to multiply 2x+4 by x+2 and combine like terms.

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

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

$$4x^{2}+8x+8=5x\left(x+2\right)$$

Use the distributive property to multiply 5x by x+2.

$$4x^{2}+8x+8=5x^{2}+10x$$

Subtract 5x^{2} from both sides.

$$4x^{2}+8x+8-5x^{2}=10x$$

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

$$-x^{2}+8x+8=10x$$

Subtract 10x from both sides.

$$-x^{2}+8x+8-10x=0$$

Combine 8x and -10x to get -2x.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-8=-8$$

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 -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -2.

$$a=2 b=-4$$

Rewrite -x^{2}-2x+8 as \left(-x^{2}+2x\right)+\left(-4x+8\right).

$$\left(-x^{2}+2x\right)+\left(-4x+8\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(-x+2\right)+4\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(x+4\right)$$

To find equation solutions, solve -x+2=0 and x+4=0.

$$x=2 x=-4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+2\right), the least common multiple of x+2,x,2.

$$2xx+\left(2x+4\right)\left(x+2\right)=5x\left(x+2\right)$$

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

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

Use the distributive property to multiply 2x+4 by x+2 and combine like terms.

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

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

$$4x^{2}+8x+8=5x\left(x+2\right)$$

Use the distributive property to multiply 5x by x+2.

$$4x^{2}+8x+8=5x^{2}+10x$$

Subtract 5x^{2} from both sides.

$$4x^{2}+8x+8-5x^{2}=10x$$

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

$$-x^{2}+8x+8=10x$$

Subtract 10x from both sides.

$$-x^{2}+8x+8-10x=0$$

Combine 8x and -10x to get -2x.

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

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

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

Square -2.

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

Multiply -4 times -1.

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

Multiply 4 times 8.

$$x=\frac{-\left(-2\right)±\sqrt{4+32}}{2\left(-1\right)}$$

Add 4 to 32.

$$x=\frac{-\left(-2\right)±\sqrt{36}}{2\left(-1\right)}$$

Take the square root of 36.

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

The opposite of -2 is 2.

$$x=\frac{2±6}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±6}{-2}$$

Now solve the equation x=\frac{2±6}{-2} when ± is plus. Add 2 to 6.

$$x=\frac{8}{-2}$$

Divide 8 by -2.

$$x=-4$$

Now solve the equation x=\frac{2±6}{-2} when ± is minus. Subtract 6 from 2.

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

Divide -4 by -2.

$$x=2$$

The equation is now solved.

$$x=-4 x=2$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+2\right), the least common multiple of x+2,x,2.

$$2xx+\left(2x+4\right)\left(x+2\right)=5x\left(x+2\right)$$

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

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

Use the distributive property to multiply 2x+4 by x+2 and combine like terms.

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

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

$$4x^{2}+8x+8=5x\left(x+2\right)$$

Use the distributive property to multiply 5x by x+2.

$$4x^{2}+8x+8=5x^{2}+10x$$

Subtract 5x^{2} from both sides.

$$4x^{2}+8x+8-5x^{2}=10x$$

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

$$-x^{2}+8x+8=10x$$

Subtract 10x from both sides.

$$-x^{2}+8x+8-10x=0$$

Combine 8x and -10x to get -2x.

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

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

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide -2 by -1.

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

Divide -8 by -1.

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

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

$$x^{2}+2x+1^{2}=8+1^{2}$$

Square 1.

$$x^{2}+2x+1=8+1$$

Add 8 to 1.

$$x^{2}+2x+1=9$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+1=3 x+1=-3$$

Subtract 1 from both sides of the equation.

$$x=2 x=-4$$