Solve 300/n-300=n+200/n+(n-200)

$$\frac { 300 } { n - 300 } = \frac { n + 200 } { n + ( n - 200 ) }$$

Solve for n

$$n=700$$

$$n=0$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values 100,300 since division by zero is not defined. Multiply both sides of the equation by 2\left(n-300\right)\left(n-100\right), the least common multiple of n-300,n+n-200.

$$\left(2n-200\right)\times 300=\left(n-300\right)\left(n+200\right)$$

Use the distributive property to multiply 2n-200 by 300.

$$600n-60000=\left(n-300\right)\left(n+200\right)$$

Use the distributive property to multiply n-300 by n+200 and combine like terms.

$$600n-60000=n^{2}-100n-60000$$

Subtract n^{2} from both sides.

$$600n-60000-n^{2}=-100n-60000$$

Add 100n to both sides.

$$600n-60000-n^{2}+100n=-60000$$

Combine 600n and 100n to get 700n.

$$700n-60000-n^{2}=-60000$$

Add 60000 to both sides.

$$700n-60000-n^{2}+60000=0$$

Add -60000 and 60000 to get 0.

$$700n-n^{2}=0$$

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.

$$-n^{2}+700n=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 700 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-700±\sqrt{700^{2}}}{2\left(-1\right)}$$

Take the square root of 700^{2}.

$$n=\frac{-700±700}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{-700±700}{-2}$$

Now solve the equation n=\frac{-700±700}{-2} when ± is plus. Add -700 to 700.

$$n=\frac{0}{-2}$$

Divide 0 by -2.

$$n=0$$

Now solve the equation n=\frac{-700±700}{-2} when ± is minus. Subtract 700 from -700.

$$n=-\frac{1400}{-2}$$

Divide -1400 by -2.

$$n=700$$

The equation is now solved.

$$n=0 n=700$$

Steps for Completing the Square

Variable n cannot be equal to any of the values 100,300 since division by zero is not defined. Multiply both sides of the equation by 2\left(n-300\right)\left(n-100\right), the least common multiple of n-300,n+n-200.

$$\left(2n-200\right)\times 300=\left(n-300\right)\left(n+200\right)$$

Use the distributive property to multiply 2n-200 by 300.

$$600n-60000=\left(n-300\right)\left(n+200\right)$$

Use the distributive property to multiply n-300 by n+200 and combine like terms.

$$600n-60000=n^{2}-100n-60000$$

Subtract n^{2} from both sides.

$$600n-60000-n^{2}=-100n-60000$$

Add 100n to both sides.

$$600n-60000-n^{2}+100n=-60000$$

Combine 600n and 100n to get 700n.

$$700n-60000-n^{2}=-60000$$

Add 60000 to both sides.

$$700n-n^{2}=-60000+60000$$

Add -60000 and 60000 to get 0.

$$700n-n^{2}=0$$

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.

$$-n^{2}+700n=0$$

Divide both sides by -1.

$$\frac{-n^{2}+700n}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$n^{2}+\frac{700}{-1}n=\frac{0}{-1}$$

Divide 700 by -1.

$$n^{2}-700n=\frac{0}{-1}$$

Divide 0 by -1.

$$n^{2}-700n=0$$

Divide -700, the coefficient of the x term, by 2 to get -350. Then add the square of -350 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-700n+\left(-350\right)^{2}=\left(-350\right)^{2}$$

Square -350.

$$n^{2}-700n+122500=122500$$

Factor n^{2}-700n+122500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-350\right)^{2}=122500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-350\right)^{2}}=\sqrt{122500}$$

Simplify.

$$n-350=350 n-350=-350$$

Add 350 to both sides of the equation.

$$n=700 n=0$$