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

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

Solve for x

$$x=-30$$

$$x=50$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

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

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

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

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

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

Subtract 7500x from both sides.

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

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

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

Divide both sides by 100.

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

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-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 x^{2}-20x-1500 as \left(x^{2}-50x\right)+\left(30x-1500\right).

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

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

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

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

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

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

$$x=50 x=-30$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

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

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

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

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

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

Subtract 7500x from both sides.

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

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

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

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.

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

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}.

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

Square -2000.

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

Multiply -4 times 100.

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

Multiply -400 times -150000.

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

Add 4000000 to 60000000.

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

Take the square root of 64000000.

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

The opposite of -2000 is 2000.

$$x=\frac{2000±8000}{2\times 100}$$

Multiply 2 times 100.

$$x=\frac{2000±8000}{200}$$

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

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

Divide 10000 by 200.

$$x=50$$

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

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

Divide -6000 by 200.

$$x=-30$$

The equation is now solved.

$$x=50 x=-30$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

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

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

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

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

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

Subtract 7500x from both sides.

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

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

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

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

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

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.

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

Divide both sides by 100.

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

Dividing by 100 undoes the multiplication by 100.

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

Divide -2000 by 100.

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

Divide 150000 by 100.

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

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.

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

Square -10.

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

Add 1500 to 100.

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

Factor x^{2}-20x+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-10=40 x-10=-40$$

Add 10 to both sides of the equation.

$$x=50 x=-30$$