Solve 1600/x+400-1600/x=2/3

$$\frac{ 1600 }{ x+400 } - \frac{ 1600 }{ x } = \frac{ 2 }{ 3 }$$

Solve for x (complex solution)

$$x=-200+200\sqrt{23}i\approx -200+959.166304663i$$

$$x=-200\sqrt{23}i-200\approx -200-959.166304663i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -400,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+400\right), the least common multiple of x+400,x,3.

$$3x\times 1600-\left(3x+1200\right)\times 1600=2x\left(x+400\right)$$

Multiply 3 and 1600 to get 4800.

$$4800x-\left(3x+1200\right)\times 1600=2x\left(x+400\right)$$

Use the distributive property to multiply 3x+1200 by 1600.

$$4800x-\left(4800x+1920000\right)=2x\left(x+400\right)$$

To find the opposite of 4800x+1920000, find the opposite of each term.

$$4800x-4800x-1920000=2x\left(x+400\right)$$

Combine 4800x and -4800x to get 0.

$$-1920000=2x\left(x+400\right)$$

Use the distributive property to multiply 2x by x+400.

$$-1920000=2x^{2}+800x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}+800x=-1920000$$

Add 1920000 to both sides.

$$2x^{2}+800x+1920000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 800 for b, and 1920000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-800±\sqrt{800^{2}-4\times 2\times 1920000}}{2\times 2}$$

Square 800.

$$x=\frac{-800±\sqrt{640000-4\times 2\times 1920000}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-800±\sqrt{640000-8\times 1920000}}{2\times 2}$$

Multiply -8 times 1920000.

$$x=\frac{-800±\sqrt{640000-15360000}}{2\times 2}$$

Add 640000 to -15360000.

$$x=\frac{-800±\sqrt{-14720000}}{2\times 2}$$

Take the square root of -14720000.

$$x=\frac{-800±800\sqrt{23}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-800±800\sqrt{23}i}{4}$$

Now solve the equation x=\frac{-800±800\sqrt{23}i}{4} when ± is plus. Add -800 to 800i\sqrt{23}.

$$x=\frac{-800+800\sqrt{23}i}{4}$$

Divide -800+800i\sqrt{23} by 4.

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

Now solve the equation x=\frac{-800±800\sqrt{23}i}{4} when ± is minus. Subtract 800i\sqrt{23} from -800.

$$x=\frac{-800\sqrt{23}i-800}{4}$$

Divide -800-800i\sqrt{23} by 4.

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

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -400,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+400\right), the least common multiple of x+400,x,3.

$$3x\times 1600-\left(3x+1200\right)\times 1600=2x\left(x+400\right)$$

Multiply 3 and 1600 to get 4800.

$$4800x-\left(3x+1200\right)\times 1600=2x\left(x+400\right)$$

Use the distributive property to multiply 3x+1200 by 1600.

$$4800x-\left(4800x+1920000\right)=2x\left(x+400\right)$$

To find the opposite of 4800x+1920000, find the opposite of each term.

$$4800x-4800x-1920000=2x\left(x+400\right)$$

Combine 4800x and -4800x to get 0.

$$-1920000=2x\left(x+400\right)$$

Use the distributive property to multiply 2x by x+400.

$$-1920000=2x^{2}+800x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}+800x=-1920000$$

Divide both sides by 2.

$$\frac{2x^{2}+800x}{2}=-\frac{1920000}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{800}{2}x=-\frac{1920000}{2}$$

Divide 800 by 2.

$$x^{2}+400x=-\frac{1920000}{2}$$

Divide -1920000 by 2.

$$x^{2}+400x=-960000$$

Divide 400, the coefficient of the x term, by 2 to get 200. Then add the square of 200 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+400x+200^{2}=-960000+200^{2}$$

Square 200.

$$x^{2}+400x+40000=-960000+40000$$

Add -960000 to 40000.

$$x^{2}+400x+40000=-920000$$

Factor x^{2}+400x+40000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+200\right)^{2}=-920000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+200\right)^{2}}=\sqrt{-920000}$$

Simplify.

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

Subtract 200 from both sides of the equation.

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