Solve P(x)=xP=80x-0.1x^2-(500+20x)

$$P ( x ) = x P = 80 x - 0.1 x ^ { 2 } - ( 500 + 20 x )$$

Solve for P

$$P=-\frac{x}{10}+60-\frac{500}{x}$$

$$x\neq 0$$

Steps for Solving Linear Equation

To find the opposite of 500+20x, find the opposite of each term.

$$xP=80x-0.1x^{2}-500-20x$$

Combine 80x and -20x to get 60x.

$$xP=60x-0.1x^{2}-500$$

The equation is in standard form.

$$xP=-\frac{x^{2}}{10}+60x-500$$

Divide both sides by x.

$$\frac{xP}{x}=\frac{-\frac{x^{2}}{10}+60x-500}{x}$$

Dividing by x undoes the multiplication by x.

$$P=\frac{-\frac{x^{2}}{10}+60x-500}{x}$$

Divide 60x-\frac{x^{2}}{10}-500 by x.

$$P=-\frac{x}{10}+60-\frac{500}{x}$$

Solve for x (complex solution)

$$x=5\sqrt{P^{2}-120P+3400}-5P+300$$

$$x=-5\sqrt{P^{2}-120P+3400}-5P+300$$

Solve for x

$$x=5\sqrt{P^{2}-120P+3400}-5P+300$$

$$x=-5\sqrt{P^{2}-120P+3400}-5P+300\text{, }P\geq 10\sqrt{2}+60\text{ or }P\leq 60-10\sqrt{2}$$