Solve 600/x-600/x-200=1/2

$$\frac { 600 } { x } - \frac { 600 } { x - 200 } = \frac { 1 } { 2 }$$

Solve for x (complex solution)

$$x=-100\sqrt{23}i+100\approx 100-479.583152331i$$

$$x=100+100\sqrt{23}i\approx 100+479.583152331i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,200 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-200\right), the least common multiple of x,x-200,2.

$$\left(2x-400\right)\times 600-2x\times 600=x\left(x-200\right)$$

Use the distributive property to multiply 2x-400 by 600.

$$1200x-240000-2x\times 600=x\left(x-200\right)$$

Multiply 2 and 600 to get 1200.

$$1200x-240000-1200x=x\left(x-200\right)$$

Use the distributive property to multiply x by x-200.

$$1200x-240000-1200x=x^{2}-200x$$

Subtract x^{2} from both sides.

$$1200x-240000-1200x-x^{2}=-200x$$

Add 200x to both sides.

$$1200x-240000-1200x-x^{2}+200x=0$$

Combine 1200x and 200x to get 1400x.

$$1400x-240000-1200x-x^{2}=0$$

Combine 1400x and -1200x to get 200x.

$$200x-240000-x^{2}=0$$

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.

$$-x^{2}+200x-240000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 200 for b, and -240000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-1\right)\left(-240000\right)}}{2\left(-1\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-1\right)\left(-240000\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-200±\sqrt{40000+4\left(-240000\right)}}{2\left(-1\right)}$$

Multiply 4 times -240000.

$$x=\frac{-200±\sqrt{40000-960000}}{2\left(-1\right)}$$

Add 40000 to -960000.

$$x=\frac{-200±\sqrt{-920000}}{2\left(-1\right)}$$

Take the square root of -920000.

$$x=\frac{-200±200\sqrt{23}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-200±200\sqrt{23}i}{-2}$$

Now solve the equation x=\frac{-200±200\sqrt{23}i}{-2} when ± is plus. Add -200 to 200i\sqrt{23}.

$$x=\frac{-200+200\sqrt{23}i}{-2}$$

Divide -200+200i\sqrt{23} by -2.

$$x=-100\sqrt{23}i+100$$

Now solve the equation x=\frac{-200±200\sqrt{23}i}{-2} when ± is minus. Subtract 200i\sqrt{23} from -200.

$$x=\frac{-200\sqrt{23}i-200}{-2}$$

Divide -200-200i\sqrt{23} by -2.

$$x=100+100\sqrt{23}i$$

The equation is now solved.

$$x=-100\sqrt{23}i+100 x=100+100\sqrt{23}i$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,200 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-200\right), the least common multiple of x,x-200,2.

$$\left(2x-400\right)\times 600-2x\times 600=x\left(x-200\right)$$

Use the distributive property to multiply 2x-400 by 600.

$$1200x-240000-2x\times 600=x\left(x-200\right)$$

Multiply 2 and 600 to get 1200.

$$1200x-240000-1200x=x\left(x-200\right)$$

Use the distributive property to multiply x by x-200.

$$1200x-240000-1200x=x^{2}-200x$$

Subtract x^{2} from both sides.

$$1200x-240000-1200x-x^{2}=-200x$$

Add 200x to both sides.

$$1200x-240000-1200x-x^{2}+200x=0$$

Combine 1200x and 200x to get 1400x.

$$1400x-240000-1200x-x^{2}=0$$

Add 240000 to both sides. Anything plus zero gives itself.

$$1400x-1200x-x^{2}=240000$$

Combine 1400x and -1200x to get 200x.

$$200x-x^{2}=240000$$

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.

$$-x^{2}+200x=240000$$

Divide both sides by -1.

$$\frac{-x^{2}+200x}{-1}=\frac{240000}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{200}{-1}x=\frac{240000}{-1}$$

Divide 200 by -1.

$$x^{2}-200x=\frac{240000}{-1}$$

Divide 240000 by -1.

$$x^{2}-200x=-240000$$

Divide -200, the coefficient of the x term, by 2 to get -100. Then add the square of -100 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-200x+\left(-100\right)^{2}=-240000+\left(-100\right)^{2}$$

Square -100.

$$x^{2}-200x+10000=-240000+10000$$

Add -240000 to 10000.

$$x^{2}-200x+10000=-230000$$

Factor x^{2}-200x+10000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-100\right)^{2}=-230000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-100\right)^{2}}=\sqrt{-230000}$$

Simplify.

$$x-100=100\sqrt{23}i x-100=-100\sqrt{23}i$$

Add 100 to both sides of the equation.

$$x=100+100\sqrt{23}i x=-100\sqrt{23}i+100$$