Solve 1=2+200-30n/n^2

$$1 = 2 + \frac { 200 - 30 n } { n ^ { 2 } }$$

Solve for n

$$n=10$$

$$n=20$$

Steps Using Factoring By Grouping

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n^{2}.

$$n^{2}=n^{2}\times 2+200-30n$$

Subtract n^{2}\times 2 from both sides.

$$n^{2}-n^{2}\times 2=200-30n$$

Combine n^{2} and -n^{2}\times 2 to get -n^{2}.

$$-n^{2}=200-30n$$

Subtract 200 from both sides.

$$-n^{2}-200=-30n$$

Add 30n to both sides.

$$-n^{2}-200+30n=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-n^{2}+30n-200=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -n^{2}+an+bn-200. To find a and b, set up a system to be solved.

$$a+b=30 ab=-\left(-200\right)=200$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 200.

$$1,200 2,100 4,50 5,40 8,25 10,20$$

Calculate the sum for each pair.

$$1+200=201 2+100=102 4+50=54 5+40=45 8+25=33 10+20=30$$

The solution is the pair that gives sum 30.

$$a=20 b=10$$

Rewrite -n^{2}+30n-200 as \left(-n^{2}+20n\right)+\left(10n-200\right).

$$\left(-n^{2}+20n\right)+\left(10n-200\right)$$

Factor out -n in the first and 10 in the second group.

$$-n\left(n-20\right)+10\left(n-20\right)$$

Factor out common term n-20 by using distributive property.

$$\left(n-20\right)\left(-n+10\right)$$

To find equation solutions, solve n-20=0 and -n+10=0.

$$n=20 n=10$$

Steps Using the Quadratic Formula

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n^{2}.

$$n^{2}=n^{2}\times 2+200-30n$$

Subtract n^{2}\times 2 from both sides.

$$n^{2}-n^{2}\times 2=200-30n$$

Combine n^{2} and -n^{2}\times 2 to get -n^{2}.

$$-n^{2}=200-30n$$

Subtract 200 from both sides.

$$-n^{2}-200=-30n$$

Add 30n to both sides.

$$-n^{2}-200+30n=0$$

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.

$$-n^{2}+30n-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 30 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-30±\sqrt{30^{2}-4\left(-1\right)\left(-200\right)}}{2\left(-1\right)}$$

Square 30.

$$n=\frac{-30±\sqrt{900-4\left(-1\right)\left(-200\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{-30±\sqrt{900+4\left(-200\right)}}{2\left(-1\right)}$$

Multiply 4 times -200.

$$n=\frac{-30±\sqrt{900-800}}{2\left(-1\right)}$$

Add 900 to -800.

$$n=\frac{-30±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$n=\frac{-30±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{-30±10}{-2}$$

Now solve the equation n=\frac{-30±10}{-2} when ± is plus. Add -30 to 10.

$$n=-\frac{20}{-2}$$

Divide -20 by -2.

$$n=10$$

Now solve the equation n=\frac{-30±10}{-2} when ± is minus. Subtract 10 from -30.

$$n=-\frac{40}{-2}$$

Divide -40 by -2.

$$n=20$$

The equation is now solved.

$$n=10 n=20$$

Steps for Completing the Square

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n^{2}.

$$n^{2}=n^{2}\times 2+200-30n$$

Subtract n^{2}\times 2 from both sides.

$$n^{2}-n^{2}\times 2=200-30n$$

Combine n^{2} and -n^{2}\times 2 to get -n^{2}.

$$-n^{2}=200-30n$$

Add 30n to both sides.

$$-n^{2}+30n=200$$

Divide both sides by -1.

$$\frac{-n^{2}+30n}{-1}=\frac{200}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$n^{2}+\frac{30}{-1}n=\frac{200}{-1}$$

Divide 30 by -1.

$$n^{2}-30n=\frac{200}{-1}$$

Divide 200 by -1.

$$n^{2}-30n=-200$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-30n+\left(-15\right)^{2}=-200+\left(-15\right)^{2}$$

Square -15.

$$n^{2}-30n+225=-200+225$$

Add -200 to 225.

$$n^{2}-30n+225=25$$

Factor n^{2}-30n+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-15\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-15\right)^{2}}=\sqrt{25}$$

Simplify.

$$n-15=5 n-15=-5$$

Add 15 to both sides of the equation.

$$n=20 n=10$$