Solve 10000=6500/1+rho+9200/(1+p)^2

$$10000 = \frac { 6500 } { 1 + \rho } + \frac { 9200 } { ( 1 + p ) ^ { 2 } }$$

Solve for ρ

$$\rho =-\frac{35p^{2}+70p-57}{4\left(25p^{2}+50p+2\right)}$$

$$p\neq -1\text{ and }p\neq \frac{\sqrt{23}}{5}-1\text{ and }p\neq -\frac{\sqrt{23}}{5}-1$$

Steps for Solving Linear Equation

Variable \rho cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(\rho +1\right)\left(p+1\right)^{2}, the least common multiple of 1+\rho ,\left(1+p\right)^{2}.

$$10000\left(\rho +1\right)\left(p+1\right)^{2}=\left(p+1\right)^{2}\times 6500+\left(\rho +1\right)\times 9200$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(p+1\right)^{2}.

$$10000\left(\rho +1\right)\left(p^{2}+2p+1\right)=\left(p+1\right)^{2}\times 6500+\left(\rho +1\right)\times 9200$$

Use the distributive property to multiply 10000 by \rho +1.

$$\left(10000\rho +10000\right)\left(p^{2}+2p+1\right)=\left(p+1\right)^{2}\times 6500+\left(\rho +1\right)\times 9200$$

Use the distributive property to multiply 10000\rho +10000 by p^{2}+2p+1.

$$10000\rho p^{2}+20000\rho p+10000\rho +10000p^{2}+20000p+10000=\left(p+1\right)^{2}\times 6500+\left(\rho +1\right)\times 9200$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(p+1\right)^{2}.

$$10000\rho p^{2}+20000\rho p+10000\rho +10000p^{2}+20000p+10000=\left(p^{2}+2p+1\right)\times 6500+\left(\rho +1\right)\times 9200$$

Use the distributive property to multiply p^{2}+2p+1 by 6500.

$$10000\rho p^{2}+20000\rho p+10000\rho +10000p^{2}+20000p+10000=6500p^{2}+13000p+6500+\left(\rho +1\right)\times 9200$$

Use the distributive property to multiply \rho +1 by 9200.

$$10000\rho p^{2}+20000\rho p+10000\rho +10000p^{2}+20000p+10000=6500p^{2}+13000p+6500+9200\rho +9200$$

Add 6500 and 9200 to get 15700.

$$10000\rho p^{2}+20000\rho p+10000\rho +10000p^{2}+20000p+10000=6500p^{2}+13000p+15700+9200\rho$$

Subtract 9200\rho from both sides.

$$10000\rho p^{2}+20000\rho p+10000\rho +10000p^{2}+20000p+10000-9200\rho =6500p^{2}+13000p+15700$$

Combine 10000\rho and -9200\rho to get 800\rho .

$$10000\rho p^{2}+20000\rho p+800\rho +10000p^{2}+20000p+10000=6500p^{2}+13000p+15700$$

Subtract 10000p^{2} from both sides.

$$10000\rho p^{2}+20000\rho p+800\rho +20000p+10000=6500p^{2}+13000p+15700-10000p^{2}$$

Combine 6500p^{2} and -10000p^{2} to get -3500p^{2}.

$$10000\rho p^{2}+20000\rho p+800\rho +20000p+10000=-3500p^{2}+13000p+15700$$

Subtract 20000p from both sides.

$$10000\rho p^{2}+20000\rho p+800\rho +10000=-3500p^{2}+13000p+15700-20000p$$

Combine 13000p and -20000p to get -7000p.

$$10000\rho p^{2}+20000\rho p+800\rho +10000=-3500p^{2}-7000p+15700$$

Subtract 10000 from both sides.

$$10000\rho p^{2}+20000\rho p+800\rho =-3500p^{2}-7000p+15700-10000$$

Subtract 10000 from 15700 to get 5700.

$$10000\rho p^{2}+20000\rho p+800\rho =-3500p^{2}-7000p+5700$$

Combine all terms containing \rho .

$$\left(10000p^{2}+20000p+800\right)\rho =-3500p^{2}-7000p+5700$$

The equation is in standard form.

$$\left(10000p^{2}+20000p+800\right)\rho =5700-7000p-3500p^{2}$$

Divide both sides by 800+20000p+10000p^{2}.

$$\frac{\left(10000p^{2}+20000p+800\right)\rho }{10000p^{2}+20000p+800}=\frac{5700-7000p-3500p^{2}}{10000p^{2}+20000p+800}$$

Dividing by 800+20000p+10000p^{2} undoes the multiplication by 800+20000p+10000p^{2}.

$$\rho =\frac{5700-7000p-3500p^{2}}{10000p^{2}+20000p+800}$$

Divide -3500p^{2}-7000p+5700 by 800+20000p+10000p^{2}.

$$\rho =\frac{57-70p-35p^{2}}{4\left(25p^{2}+50p+2\right)}$$

Variable \rho cannot be equal to -1.

$$\rho =\frac{57-70p-35p^{2}}{4\left(25p^{2}+50p+2\right)}\text{, }\rho \neq -1$$

Solve for p (complex solution)

$$p=-\frac{2i\left(-20\rho -7\right)^{-\frac{1}{2}}\sqrt{115\rho +115}}{5}-1$$

$$p=\frac{2i\left(-20\rho -7\right)^{-\frac{1}{2}}\sqrt{115\rho +115}}{5}-1\text{, }\rho \neq -1\text{ and }\rho \neq -\frac{7}{20}$$

Solve for p

$$\left\{\begin{matrix}p=\frac{-2\sqrt{115\left(\rho +1\right)}-5\sqrt{20\rho +7}}{5\sqrt{20\rho +7}}\text{; }p=\frac{2\sqrt{115\left(\rho +1\right)}-5\sqrt{20\rho +7}}{5\sqrt{20\rho +7}}\text{, }&\rho >-\frac{7}{20}\\p=\frac{-2\sqrt{\frac{115\left(\rho +1\right)}{20\rho +7}}-5}{5}\text{; }p=\frac{2\sqrt{\frac{115\left(\rho +1\right)}{20\rho +7}}-5}{5}\text{, }&\rho <-1\end{matrix}\right.$$