Solve 6000/x-400-6000/x=301/60

$$\frac{ 6000 }{ x-400 } - \frac{ 6000 }{ x } = 30 \frac{ 1 }{ 60 }$$

Solve for x

$$x=\frac{200\sqrt{9727201}}{1801}+200\approx 546.346041169$$

$$x=-\frac{200\sqrt{9727201}}{1801}+200\approx -146.346041169$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,400 since division by zero is not defined. Multiply both sides of the equation by 60x\left(x-400\right), the least common multiple of x-400,x,60.

$$60x\times 6000-\left(60x-24000\right)\times 6000=x\left(x-400\right)\left(30\times 60+1\right)$$

Multiply 60 and 6000 to get 360000.

$$360000x-\left(60x-24000\right)\times 6000=x\left(x-400\right)\left(30\times 60+1\right)$$

Use the distributive property to multiply 60x-24000 by 6000.

$$360000x-\left(360000x-144000000\right)=x\left(x-400\right)\left(30\times 60+1\right)$$

To find the opposite of 360000x-144000000, find the opposite of each term.

$$360000x-360000x+144000000=x\left(x-400\right)\left(30\times 60+1\right)$$

Combine 360000x and -360000x to get 0.

$$144000000=x\left(x-400\right)\left(30\times 60+1\right)$$

Multiply 30 and 60 to get 1800.

$$144000000=x\left(x-400\right)\left(1800+1\right)$$

Add 1800 and 1 to get 1801.

$$144000000=x\left(x-400\right)\times 1801$$

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

$$144000000=\left(x^{2}-400x\right)\times 1801$$

Use the distributive property to multiply x^{2}-400x by 1801.

$$144000000=1801x^{2}-720400x$$

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

$$1801x^{2}-720400x=144000000$$

Subtract 144000000 from both sides.

$$1801x^{2}-720400x-144000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1801 for a, -720400 for b, and -144000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-720400\right)±\sqrt{\left(-720400\right)^{2}-4\times 1801\left(-144000000\right)}}{2\times 1801}$$

Square -720400.

$$x=\frac{-\left(-720400\right)±\sqrt{518976160000-4\times 1801\left(-144000000\right)}}{2\times 1801}$$

Multiply -4 times 1801.

$$x=\frac{-\left(-720400\right)±\sqrt{518976160000-7204\left(-144000000\right)}}{2\times 1801}$$

Multiply -7204 times -144000000.

$$x=\frac{-\left(-720400\right)±\sqrt{518976160000+1037376000000}}{2\times 1801}$$

Add 518976160000 to 1037376000000.

$$x=\frac{-\left(-720400\right)±\sqrt{1556352160000}}{2\times 1801}$$

Take the square root of 1556352160000.

$$x=\frac{-\left(-720400\right)±400\sqrt{9727201}}{2\times 1801}$$

The opposite of -720400 is 720400.

$$x=\frac{720400±400\sqrt{9727201}}{2\times 1801}$$

Multiply 2 times 1801.

$$x=\frac{720400±400\sqrt{9727201}}{3602}$$

Now solve the equation x=\frac{720400±400\sqrt{9727201}}{3602} when ± is plus. Add 720400 to 400\sqrt{9727201}.

$$x=\frac{400\sqrt{9727201}+720400}{3602}$$

Divide 720400+400\sqrt{9727201} by 3602.

$$x=\frac{200\sqrt{9727201}}{1801}+200$$

Now solve the equation x=\frac{720400±400\sqrt{9727201}}{3602} when ± is minus. Subtract 400\sqrt{9727201} from 720400.

$$x=\frac{720400-400\sqrt{9727201}}{3602}$$

Divide 720400-400\sqrt{9727201} by 3602.

$$x=-\frac{200\sqrt{9727201}}{1801}+200$$

The equation is now solved.

$$x=\frac{200\sqrt{9727201}}{1801}+200 x=-\frac{200\sqrt{9727201}}{1801}+200$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,400 since division by zero is not defined. Multiply both sides of the equation by 60x\left(x-400\right), the least common multiple of x-400,x,60.

$$60x\times 6000-\left(60x-24000\right)\times 6000=x\left(x-400\right)\left(30\times 60+1\right)$$

Multiply 60 and 6000 to get 360000.

$$360000x-\left(60x-24000\right)\times 6000=x\left(x-400\right)\left(30\times 60+1\right)$$

Use the distributive property to multiply 60x-24000 by 6000.

$$360000x-\left(360000x-144000000\right)=x\left(x-400\right)\left(30\times 60+1\right)$$

To find the opposite of 360000x-144000000, find the opposite of each term.

$$360000x-360000x+144000000=x\left(x-400\right)\left(30\times 60+1\right)$$

Combine 360000x and -360000x to get 0.

$$144000000=x\left(x-400\right)\left(30\times 60+1\right)$$

Multiply 30 and 60 to get 1800.

$$144000000=x\left(x-400\right)\left(1800+1\right)$$

Add 1800 and 1 to get 1801.

$$144000000=x\left(x-400\right)\times 1801$$

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

$$144000000=\left(x^{2}-400x\right)\times 1801$$

Use the distributive property to multiply x^{2}-400x by 1801.

$$144000000=1801x^{2}-720400x$$

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

$$1801x^{2}-720400x=144000000$$

Divide both sides by 1801.

$$\frac{1801x^{2}-720400x}{1801}=\frac{144000000}{1801}$$

Dividing by 1801 undoes the multiplication by 1801.

$$x^{2}+\left(-\frac{720400}{1801}\right)x=\frac{144000000}{1801}$$

Divide -720400 by 1801.

$$x^{2}-400x=\frac{144000000}{1801}$$

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.

$$x^{2}-400x+\left(-200\right)^{2}=\frac{144000000}{1801}+\left(-200\right)^{2}$$

Square -200.

$$x^{2}-400x+40000=\frac{144000000}{1801}+40000$$

Add \frac{144000000}{1801} to 40000.

$$x^{2}-400x+40000=\frac{216040000}{1801}$$

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}.

$$\left(x-200\right)^{2}=\frac{216040000}{1801}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-200\right)^{2}}=\sqrt{\frac{216040000}{1801}}$$

Simplify.

$$x-200=\frac{200\sqrt{9727201}}{1801} x-200=-\frac{200\sqrt{9727201}}{1801}$$

Add 200 to both sides of the equation.

$$x=\frac{200\sqrt{9727201}}{1801}+200 x=-\frac{200\sqrt{9727201}}{1801}+200$$