Solve 20/x=200/(x+20)+1/2

$$\frac { 20 } { x } = \frac { 200 } { ( x + 20 ) } + \frac { 1 } { 2 }$$

Solve for x

$$x=30\sqrt{41}-190\approx 2.093727123$$

$$x=-30\sqrt{41}-190\approx -382.093727123$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -20,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+20\right), the least common multiple of x,x+20,2.

$$\left(2x+40\right)\times 20=2x\times 200+2x\left(x+20\right)\times \frac{1}{2}$$

Use the distributive property to multiply 2x+40 by 20.

$$40x+800=2x\times 200+2x\left(x+20\right)\times \frac{1}{2}$$

Multiply 2 and 200 to get 400.

$$40x+800=400x+2x\left(x+20\right)\times \frac{1}{2}$$

Multiply 2 and \frac{1}{2} to get 1.

$$40x+800=400x+x\left(x+20\right)$$

Use the distributive property to multiply x by x+20.

$$40x+800=400x+x^{2}+20x$$

Combine 400x and 20x to get 420x.

$$40x+800=420x+x^{2}$$

Subtract 420x from both sides.

$$40x+800-420x=x^{2}$$

Combine 40x and -420x to get -380x.

$$-380x+800=x^{2}$$

Subtract x^{2} from both sides.

$$-380x+800-x^{2}=0$$

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.

$$-x^{2}-380x+800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -380 for b, and 800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-380\right)±\sqrt{\left(-380\right)^{2}-4\left(-1\right)\times 800}}{2\left(-1\right)}$$

Square -380.

$$x=\frac{-\left(-380\right)±\sqrt{144400-4\left(-1\right)\times 800}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-380\right)±\sqrt{144400+4\times 800}}{2\left(-1\right)}$$

Multiply 4 times 800.

$$x=\frac{-\left(-380\right)±\sqrt{144400+3200}}{2\left(-1\right)}$$

Add 144400 to 3200.

$$x=\frac{-\left(-380\right)±\sqrt{147600}}{2\left(-1\right)}$$

Take the square root of 147600.

$$x=\frac{-\left(-380\right)±60\sqrt{41}}{2\left(-1\right)}$$

The opposite of -380 is 380.

$$x=\frac{380±60\sqrt{41}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{380±60\sqrt{41}}{-2}$$

Now solve the equation x=\frac{380±60\sqrt{41}}{-2} when ± is plus. Add 380 to 60\sqrt{41}.

$$x=\frac{60\sqrt{41}+380}{-2}$$

Divide 380+60\sqrt{41} by -2.

$$x=-30\sqrt{41}-190$$

Now solve the equation x=\frac{380±60\sqrt{41}}{-2} when ± is minus. Subtract 60\sqrt{41} from 380.

$$x=\frac{380-60\sqrt{41}}{-2}$$

Divide 380-60\sqrt{41} by -2.

$$x=30\sqrt{41}-190$$

The equation is now solved.

$$x=-30\sqrt{41}-190 x=30\sqrt{41}-190$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -20,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+20\right), the least common multiple of x,x+20,2.

$$\left(2x+40\right)\times 20=2x\times 200+2x\left(x+20\right)\times \frac{1}{2}$$

Use the distributive property to multiply 2x+40 by 20.

$$40x+800=2x\times 200+2x\left(x+20\right)\times \frac{1}{2}$$

Multiply 2 and 200 to get 400.

$$40x+800=400x+2x\left(x+20\right)\times \frac{1}{2}$$

Multiply 2 and \frac{1}{2} to get 1.

$$40x+800=400x+x\left(x+20\right)$$

Use the distributive property to multiply x by x+20.

$$40x+800=400x+x^{2}+20x$$

Combine 400x and 20x to get 420x.

$$40x+800=420x+x^{2}$$

Subtract 420x from both sides.

$$40x+800-420x=x^{2}$$

Combine 40x and -420x to get -380x.

$$-380x+800=x^{2}$$

Subtract x^{2} from both sides.

$$-380x+800-x^{2}=0$$

Subtract 800 from both sides. Anything subtracted from zero gives its negation.

$$-380x-x^{2}=-800$$

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.

$$-x^{2}-380x=-800$$

Divide both sides by -1.

$$\frac{-x^{2}-380x}{-1}=-\frac{800}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{380}{-1}\right)x=-\frac{800}{-1}$$

Divide -380 by -1.

$$x^{2}+380x=-\frac{800}{-1}$$

Divide -800 by -1.

$$x^{2}+380x=800$$

Divide 380, the coefficient of the x term, by 2 to get 190. Then add the square of 190 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+380x+190^{2}=800+190^{2}$$

Square 190.

$$x^{2}+380x+36100=800+36100$$

Add 800 to 36100.

$$x^{2}+380x+36100=36900$$

Factor x^{2}+380x+36100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+190\right)^{2}=36900$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+190\right)^{2}}=\sqrt{36900}$$

Simplify.

$$x+190=30\sqrt{41} x+190=-30\sqrt{41}$$

Subtract 190 from both sides of the equation.

$$x=30\sqrt{41}-190 x=-30\sqrt{41}-190$$