Solve 80/x-2=80/x+2

$$\frac { 80 } { x } - 2 = \frac { 80 } { x + 2 }$$

Solve for x

$$x=-10$$

$$x=8$$

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

$$\left(x+2\right)\times 80+x\left(x+2\right)\left(-2\right)=x\times 80$$

Use the distributive property to multiply x+2 by 80.

$$80x+160+x\left(x+2\right)\left(-2\right)=x\times 80$$

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

$$80x+160+\left(x^{2}+2x\right)\left(-2\right)=x\times 80$$

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

$$80x+160-2x^{2}-4x=x\times 80$$

Combine 80x and -4x to get 76x.

$$76x+160-2x^{2}=x\times 80$$

Subtract x\times 80 from both sides.

$$76x+160-2x^{2}-x\times 80=0$$

Combine 76x and -x\times 80 to get -4x.

$$-4x+160-2x^{2}=0$$

Divide both sides by 2.

$$-2x+80-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}-2x+80=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+80. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-80=-80$$

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 -80.

$$1,-80 2,-40 4,-20 5,-16 8,-10$$

Calculate the sum for each pair.

$$1-80=-79 2-40=-38 4-20=-16 5-16=-11 8-10=-2$$

The solution is the pair that gives sum -2.

$$a=8 b=-10$$

Rewrite -x^{2}-2x+80 as \left(-x^{2}+8x\right)+\left(-10x+80\right).

$$\left(-x^{2}+8x\right)+\left(-10x+80\right)$$

Factor out x in the first and 10 in the second group.

$$x\left(-x+8\right)+10\left(-x+8\right)$$

Factor out common term -x+8 by using distributive property.

$$\left(-x+8\right)\left(x+10\right)$$

To find equation solutions, solve -x+8=0 and x+10=0.

$$x=8 x=-10$$

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

$$\left(x+2\right)\times 80+x\left(x+2\right)\left(-2\right)=x\times 80$$

Use the distributive property to multiply x+2 by 80.

$$80x+160+x\left(x+2\right)\left(-2\right)=x\times 80$$

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

$$80x+160+\left(x^{2}+2x\right)\left(-2\right)=x\times 80$$

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

$$80x+160-2x^{2}-4x=x\times 80$$

Combine 80x and -4x to get 76x.

$$76x+160-2x^{2}=x\times 80$$

Subtract x\times 80 from both sides.

$$76x+160-2x^{2}-x\times 80=0$$

Combine 76x and -x\times 80 to get -4x.

$$-4x+160-2x^{2}=0$$

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.

$$-2x^{2}-4x+160=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -4 for b, and 160 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-2\right)\times 160}}{2\left(-2\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-2\right)\times 160}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-4\right)±\sqrt{16+8\times 160}}{2\left(-2\right)}$$

Multiply 8 times 160.

$$x=\frac{-\left(-4\right)±\sqrt{16+1280}}{2\left(-2\right)}$$

Add 16 to 1280.

$$x=\frac{-\left(-4\right)±\sqrt{1296}}{2\left(-2\right)}$$

Take the square root of 1296.

$$x=\frac{-\left(-4\right)±36}{2\left(-2\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±36}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{4±36}{-4}$$

Now solve the equation x=\frac{4±36}{-4} when ± is plus. Add 4 to 36.

$$x=\frac{40}{-4}$$

Divide 40 by -4.

$$x=-10$$

Now solve the equation x=\frac{4±36}{-4} when ± is minus. Subtract 36 from 4.

$$x=-\frac{32}{-4}$$

Divide -32 by -4.

$$x=8$$

The equation is now solved.

$$x=-10 x=8$$

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

$$\left(x+2\right)\times 80+x\left(x+2\right)\left(-2\right)=x\times 80$$

Use the distributive property to multiply x+2 by 80.

$$80x+160+x\left(x+2\right)\left(-2\right)=x\times 80$$

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

$$80x+160+\left(x^{2}+2x\right)\left(-2\right)=x\times 80$$

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

$$80x+160-2x^{2}-4x=x\times 80$$

Combine 80x and -4x to get 76x.

$$76x+160-2x^{2}=x\times 80$$

Subtract x\times 80 from both sides.

$$76x+160-2x^{2}-x\times 80=0$$

Combine 76x and -x\times 80 to get -4x.

$$-4x+160-2x^{2}=0$$

Subtract 160 from both sides. Anything subtracted from zero gives its negation.

$$-4x-2x^{2}=-160$$

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.

$$-2x^{2}-4x=-160$$

Divide both sides by -2.

$$\frac{-2x^{2}-4x}{-2}=-\frac{160}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{4}{-2}\right)x=-\frac{160}{-2}$$

Divide -4 by -2.

$$x^{2}+2x=-\frac{160}{-2}$$

Divide -160 by -2.

$$x^{2}+2x=80$$

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.

$$x^{2}+2x+1^{2}=80+1^{2}$$

Square 1.

$$x^{2}+2x+1=80+1$$

Add 80 to 1.

$$x^{2}+2x+1=81$$

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}.

$$\left(x+1\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{81}$$

Simplify.

$$x+1=9 x+1=-9$$

Subtract 1 from both sides of the equation.

$$x=8 x=-10$$