Solve 160/x-160/x+4=2

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

Solve for x

$$x=-20$$

$$x=16$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+4\right), the least common multiple of x,x+4.

$$\left(x+4\right)\times 160-x\times 160=2x\left(x+4\right)$$

Use the distributive property to multiply x+4 by 160.

$$160x+640-x\times 160=2x\left(x+4\right)$$

Use the distributive property to multiply 2x by x+4.

$$160x+640-x\times 160=2x^{2}+8x$$

Subtract 2x^{2} from both sides.

$$160x+640-x\times 160-2x^{2}=8x$$

Subtract 8x from both sides.

$$160x+640-x\times 160-2x^{2}-8x=0$$

Combine 160x and -8x to get 152x.

$$152x+640-x\times 160-2x^{2}=0$$

Multiply -1 and 160 to get -160.

$$152x+640-160x-2x^{2}=0$$

Combine 152x and -160x to get -8x.

$$-8x+640-2x^{2}=0$$

Divide both sides by 2.

$$-4x+320-x^{2}=0$$

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

$$-x^{2}-4x+320=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+320. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-320=-320$$

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 -320.

$$1,-320 2,-160 4,-80 5,-64 8,-40 10,-32 16,-20$$

Calculate the sum for each pair.

$$1-320=-319 2-160=-158 4-80=-76 5-64=-59 8-40=-32 10-32=-22 16-20=-4$$

The solution is the pair that gives sum -4.

$$a=16 b=-20$$

Rewrite -x^{2}-4x+320 as \left(-x^{2}+16x\right)+\left(-20x+320\right).

$$\left(-x^{2}+16x\right)+\left(-20x+320\right)$$

Factor out x in the first and 20 in the second group.

$$x\left(-x+16\right)+20\left(-x+16\right)$$

Factor out common term -x+16 by using distributive property.

$$\left(-x+16\right)\left(x+20\right)$$

To find equation solutions, solve -x+16=0 and x+20=0.

$$x=16 x=-20$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+4\right), the least common multiple of x,x+4.

$$\left(x+4\right)\times 160-x\times 160=2x\left(x+4\right)$$

Use the distributive property to multiply x+4 by 160.

$$160x+640-x\times 160=2x\left(x+4\right)$$

Use the distributive property to multiply 2x by x+4.

$$160x+640-x\times 160=2x^{2}+8x$$

Subtract 2x^{2} from both sides.

$$160x+640-x\times 160-2x^{2}=8x$$

Subtract 8x from both sides.

$$160x+640-x\times 160-2x^{2}-8x=0$$

Combine 160x and -8x to get 152x.

$$152x+640-x\times 160-2x^{2}=0$$

Multiply -1 and 160 to get -160.

$$152x+640-160x-2x^{2}=0$$

Combine 152x and -160x to get -8x.

$$-8x+640-2x^{2}=0$$

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.

$$-2x^{2}-8x+640=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -8 for b, and 640 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-2\right)\times 640}}{2\left(-2\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-2\right)\times 640}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-8\right)±\sqrt{64+8\times 640}}{2\left(-2\right)}$$

Multiply 8 times 640.

$$x=\frac{-\left(-8\right)±\sqrt{64+5120}}{2\left(-2\right)}$$

Add 64 to 5120.

$$x=\frac{-\left(-8\right)±\sqrt{5184}}{2\left(-2\right)}$$

Take the square root of 5184.

$$x=\frac{-\left(-8\right)±72}{2\left(-2\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±72}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{8±72}{-4}$$

Now solve the equation x=\frac{8±72}{-4} when ± is plus. Add 8 to 72.

$$x=\frac{80}{-4}$$

Divide 80 by -4.

$$x=-20$$

Now solve the equation x=\frac{8±72}{-4} when ± is minus. Subtract 72 from 8.

$$x=-\frac{64}{-4}$$

Divide -64 by -4.

$$x=16$$

The equation is now solved.

$$x=-20 x=16$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+4\right), the least common multiple of x,x+4.

$$\left(x+4\right)\times 160-x\times 160=2x\left(x+4\right)$$

Use the distributive property to multiply x+4 by 160.

$$160x+640-x\times 160=2x\left(x+4\right)$$

Use the distributive property to multiply 2x by x+4.

$$160x+640-x\times 160=2x^{2}+8x$$

Subtract 2x^{2} from both sides.

$$160x+640-x\times 160-2x^{2}=8x$$

Subtract 8x from both sides.

$$160x+640-x\times 160-2x^{2}-8x=0$$

Combine 160x and -8x to get 152x.

$$152x+640-x\times 160-2x^{2}=0$$

Subtract 640 from both sides. Anything subtracted from zero gives its negation.

$$152x-x\times 160-2x^{2}=-640$$

Multiply -1 and 160 to get -160.

$$152x-160x-2x^{2}=-640$$

Combine 152x and -160x to get -8x.

$$-8x-2x^{2}=-640$$

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.

$$-2x^{2}-8x=-640$$

Divide both sides by -2.

$$\frac{-2x^{2}-8x}{-2}=-\frac{640}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{8}{-2}\right)x=-\frac{640}{-2}$$

Divide -8 by -2.

$$x^{2}+4x=-\frac{640}{-2}$$

Divide -640 by -2.

$$x^{2}+4x=320$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=320+2^{2}$$

Square 2.

$$x^{2}+4x+4=320+4$$

Add 320 to 4.

$$x^{2}+4x+4=324$$

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=324$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{324}$$

Simplify.

$$x+2=18 x+2=-18$$

Subtract 2 from both sides of the equation.

$$x=16 x=-20$$