Solve (n-20)(7500+100n/n)=7500

$$( n - 20 ) ( \frac { 7500 + 100 n } { n } ) = 7500$$

Solve for n

$$n=-30$$

$$n=50$$

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.

$$\left(n-20\right)\left(7500+100n\right)=7500n$$

Apply the distributive property by multiplying each term of n-20 by each term of 7500+100n.

$$7500n+100n^{2}-150000-2000n=7500n$$

Combine 7500n and -2000n to get 5500n.

$$5500n+100n^{2}-150000=7500n$$

Subtract 7500n from both sides.

$$5500n+100n^{2}-150000-7500n=0$$

Combine 5500n and -7500n to get -2000n.

$$-2000n+100n^{2}-150000=0$$

Divide both sides by 100.

$$-20n+n^{2}-1500=0$$

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

$$n^{2}-20n-1500=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as n^{2}+an+bn-1500. To find a and b, set up a system to be solved.

$$a+b=-20 ab=1\left(-1500\right)=-1500$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -1500.

$$1,-1500 2,-750 3,-500 4,-375 5,-300 6,-250 10,-150 12,-125 15,-100 20,-75 25,-60 30,-50$$

Calculate the sum for each pair.

$$1-1500=-1499 2-750=-748 3-500=-497 4-375=-371 5-300=-295 6-250=-244 10-150=-140 12-125=-113 15-100=-85 20-75=-55 25-60=-35 30-50=-20$$

The solution is the pair that gives sum -20.

$$a=-50 b=30$$

Rewrite n^{2}-20n-1500 as \left(n^{2}-50n\right)+\left(30n-1500\right).

$$\left(n^{2}-50n\right)+\left(30n-1500\right)$$

Factor out n in the first and 30 in the second group.

$$n\left(n-50\right)+30\left(n-50\right)$$

Factor out common term n-50 by using distributive property.

$$\left(n-50\right)\left(n+30\right)$$

To find equation solutions, solve n-50=0 and n+30=0.

$$n=50 n=-30$$

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.

$$\left(n-20\right)\left(7500+100n\right)=7500n$$

Apply the distributive property by multiplying each term of n-20 by each term of 7500+100n.

$$7500n+100n^{2}-150000-2000n=7500n$$

Combine 7500n and -2000n to get 5500n.

$$5500n+100n^{2}-150000=7500n$$

Subtract 7500n from both sides.

$$5500n+100n^{2}-150000-7500n=0$$

Combine 5500n and -7500n to get -2000n.

$$-2000n+100n^{2}-150000=0$$

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.

$$100n^{2}-2000n-150000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, -2000 for b, and -150000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-2000\right)±\sqrt{\left(-2000\right)^{2}-4\times 100\left(-150000\right)}}{2\times 100}$$

Square -2000.

$$n=\frac{-\left(-2000\right)±\sqrt{4000000-4\times 100\left(-150000\right)}}{2\times 100}$$

Multiply -4 times 100.

$$n=\frac{-\left(-2000\right)±\sqrt{4000000-400\left(-150000\right)}}{2\times 100}$$

Multiply -400 times -150000.

$$n=\frac{-\left(-2000\right)±\sqrt{4000000+60000000}}{2\times 100}$$

Add 4000000 to 60000000.

$$n=\frac{-\left(-2000\right)±\sqrt{64000000}}{2\times 100}$$

Take the square root of 64000000.

$$n=\frac{-\left(-2000\right)±8000}{2\times 100}$$

The opposite of -2000 is 2000.

$$n=\frac{2000±8000}{2\times 100}$$

Multiply 2 times 100.

$$n=\frac{2000±8000}{200}$$

Now solve the equation n=\frac{2000±8000}{200} when ± is plus. Add 2000 to 8000.

$$n=\frac{10000}{200}$$

Divide 10000 by 200.

$$n=50$$

Now solve the equation n=\frac{2000±8000}{200} when ± is minus. Subtract 8000 from 2000.

$$n=-\frac{6000}{200}$$

Divide -6000 by 200.

$$n=-30$$

The equation is now solved.

$$n=50 n=-30$$

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.

$$\left(n-20\right)\left(7500+100n\right)=7500n$$

Apply the distributive property by multiplying each term of n-20 by each term of 7500+100n.

$$7500n+100n^{2}-150000-2000n=7500n$$

Combine 7500n and -2000n to get 5500n.

$$5500n+100n^{2}-150000=7500n$$

Subtract 7500n from both sides.

$$5500n+100n^{2}-150000-7500n=0$$

Combine 5500n and -7500n to get -2000n.

$$-2000n+100n^{2}-150000=0$$

Add 150000 to both sides. Anything plus zero gives itself.

$$-2000n+100n^{2}=150000$$

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.

$$100n^{2}-2000n=150000$$

Divide both sides by 100.

$$\frac{100n^{2}-2000n}{100}=\frac{150000}{100}$$

Dividing by 100 undoes the multiplication by 100.

$$n^{2}+\left(-\frac{2000}{100}\right)n=\frac{150000}{100}$$

Divide -2000 by 100.

$$n^{2}-20n=\frac{150000}{100}$$

Divide 150000 by 100.

$$n^{2}-20n=1500$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-20n+\left(-10\right)^{2}=1500+\left(-10\right)^{2}$$

Square -10.

$$n^{2}-20n+100=1500+100$$

Add 1500 to 100.

$$n^{2}-20n+100=1600$$

Factor n^{2}-20n+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-10\right)^{2}=1600$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-10\right)^{2}}=\sqrt{1600}$$

Simplify.

$$n-10=40 n-10=-40$$

Add 10 to both sides of the equation.

$$n=50 n=-30$$