Solve NPV=550,000*500,000/1+r*500,000/(1+r)^2*500,000/(1+r)^3-1,000,000/(1+r)^4

$$N P V = 550,000 \cdot \frac { 500,000 } { 1 + r } \cdot \frac { 500,000 } { ( 1 + r ) ^ { 2 } } \cdot \frac { 500,000 } { ( 1 + r ) ^ { 3 } } - \frac { 1,000,000 } { ( 1 + r ) ^ { 4 } }$$

Solve for N

$$\left\{\begin{matrix}N=-\frac{1000000\left(r^{2}+2r-68749999999999999\right)}{PV\left(r+1\right)^{6}}\text{, }&r\neq -1\text{ and }V\neq 0\text{ and }P\neq 0\\N\in \mathrm{R}\text{, }&\left(r=-25000000\sqrt{110}-1\text{ and }V=0\right)\text{ or }\left(r=25000000\sqrt{110}-1\text{ and }V=0\right)\text{ or }\left(P=0\text{ and }V\neq 0\text{ and }r=25000000\sqrt{110}-1\right)\text{ or }\left(P=0\text{ and }V\neq 0\text{ and }r=-25000000\sqrt{110}-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(r+1\right)^{4}, the least common multiple of 1+r,\left(1+r\right)^{2},\left(1+r\right)^{3},\left(1+r\right)^{4}.

$$NPV\left(r+1\right)^{4}=550000\left(r+1\right)^{3}\times 500000\times \frac{500000}{\left(1+r\right)^{2}}\times \frac{500000}{\left(1+r\right)^{3}}-1000000$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(r+1\right)^{3}.

$$NPV\left(r+1\right)^{4}=550000\left(r^{3}+3r^{2}+3r+1\right)\times 500000\times \frac{500000}{\left(1+r\right)^{2}}\times \frac{500000}{\left(1+r\right)^{3}}-1000000$$

Multiply 550000 and 500000 to get 275000000000.

$$NPV\left(r+1\right)^{4}=275000000000\left(r^{3}+3r^{2}+3r+1\right)\times \frac{500000}{\left(1+r\right)^{2}}\times \frac{500000}{\left(1+r\right)^{3}}-1000000$$

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

$$NPV\left(r+1\right)^{4}=275000000000\left(r^{3}+3r^{2}+3r+1\right)\times \frac{500000}{1+2r+r^{2}}\times \frac{500000}{\left(1+r\right)^{3}}-1000000$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(1+r\right)^{3}.

$$NPV\left(r+1\right)^{4}=275000000000\left(r^{3}+3r^{2}+3r+1\right)\times \frac{500000}{1+2r+r^{2}}\times \frac{500000}{1+3r+3r^{2}+r^{3}}-1000000$$

Express 275000000000\times \frac{500000}{1+2r+r^{2}} as a single fraction.

$$NPV\left(r+1\right)^{4}=\frac{275000000000\times 500000}{1+2r+r^{2}}\left(r^{3}+3r^{2}+3r+1\right)\times \frac{500000}{1+3r+3r^{2}+r^{3}}-1000000$$

Multiply \frac{275000000000\times 500000}{1+2r+r^{2}} times \frac{500000}{1+3r+3r^{2}+r^{3}} by multiplying numerator times numerator and denominator times denominator.

$$NPV\left(r+1\right)^{4}=\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}\left(r^{3}+3r^{2}+3r+1\right)-1000000$$

Use the distributive property to multiply \frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} by r^{3}+3r^{2}+3r+1.

$$NPV\left(r+1\right)^{4}=\frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{3}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Calculate 500000 to the power of 2 and get 250000000000.

$$NPV\left(r+1\right)^{4}=\frac{275000000000\times 250000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{3}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Multiply 275000000000 and 250000000000 to get 68750000000000000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{3}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Express \frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{3} as a single fraction.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Calculate 500000 to the power of 2 and get 250000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+3\times \frac{275000000000\times 250000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Multiply 275000000000 and 250000000000 to get 68750000000000000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+3\times \frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Express 3\times \frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} as a single fraction.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Express \frac{3\times 68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2} as a single fraction.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Calculate 500000 to the power of 2 and get 250000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+3\times \frac{275000000000\times 250000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Multiply 275000000000 and 250000000000 to get 68750000000000000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+3\times \frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Express 3\times \frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} as a single fraction.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Express \frac{3\times 68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r as a single fraction.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Multiply 275000000000 and 500000 to get 137500000000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{137500000000000000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Multiply 137500000000000000 and 500000 to get 68750000000000000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Since \frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} and \frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} have the same denominator, add them by adding their numerators.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}+3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Do the multiplications in 68750000000000000000000r^{3}+3\times 68750000000000000000000r^{2}.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Since \frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} and \frac{3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} have the same denominator, add them by adding their numerators.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}+3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Do the multiplications in 68750000000000000000000r^{3}+206250000000000000000000r^{2}+3\times 68750000000000000000000r.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}+206250000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Since \frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}+206250000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} and \frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} have the same denominator, add them by adding their numerators.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}+206250000000000000000000r+68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Factor the expressions that are not already factored in \frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}+206250000000000000000000r+68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000\left(r+1\right)^{3}}{\left(r+1\right)^{2}\left(r+1\right)^{3}}-1000000$$

Cancel out \left(r+1\right)\left(r+1\right)^{2} in both numerator and denominator.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000}{\left(r+1\right)^{2}}-1000000$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1000000 times \frac{\left(r+1\right)^{2}}{\left(r+1\right)^{2}}.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000}{\left(r+1\right)^{2}}-\frac{1000000\left(r+1\right)^{2}}{\left(r+1\right)^{2}}$$

Since \frac{68750000000000000000000}{\left(r+1\right)^{2}} and \frac{1000000\left(r+1\right)^{2}}{\left(r+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000-1000000\left(r+1\right)^{2}}{\left(r+1\right)^{2}}$$

Do the multiplications in 68750000000000000000000-1000000\left(r+1\right)^{2}.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000-1000000r^{2}-2000000r-1000000}{\left(r+1\right)^{2}}$$

Combine like terms in 68750000000000000000000-1000000r^{2}-2000000r-1000000.

$$NPV\left(r+1\right)^{4}=\frac{68749999999999999000000-1000000r^{2}-2000000r}{\left(r+1\right)^{2}}$$

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

$$NPV\left(r+1\right)^{4}=\frac{68749999999999999000000-1000000r^{2}-2000000r}{r^{2}+2r+1}$$

Multiply both sides of the equation by \left(r+1\right)^{2}.

$$NPV\left(r+1\right)^{4}\left(r+1\right)^{2}=68749999999999999000000-1000000r^{2}-2000000r$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$NPV\left(r+1\right)^{6}=68749999999999999000000-1000000r^{2}-2000000r$$

The equation is in standard form.

$$PV\left(r+1\right)^{6}N=68749999999999999000000-2000000r-1000000r^{2}$$

Divide both sides by PV\left(r+1\right)^{6}.

$$\frac{PV\left(r+1\right)^{6}N}{PV\left(r+1\right)^{6}}=\frac{68749999999999999000000-2000000r-1000000r^{2}}{PV\left(r+1\right)^{6}}$$

Dividing by PV\left(r+1\right)^{6} undoes the multiplication by PV\left(r+1\right)^{6}.

$$N=\frac{68749999999999999000000-2000000r-1000000r^{2}}{PV\left(r+1\right)^{6}}$$

Divide 68749999999999999000000-1000000r^{2}-2000000r by PV\left(r+1\right)^{6}.

$$N=\frac{1000000\left(68749999999999999-2r-r^{2}\right)}{PV\left(r+1\right)^{6}}$$

Solve for P

$$\left\{\begin{matrix}P=-\frac{1000000\left(r^{2}+2r-68749999999999999\right)}{NV\left(r+1\right)^{6}}\text{, }&r\neq -1\text{ and }V\neq 0\text{ and }N\neq 0\\P\in \mathrm{R}\text{, }&\left(r=-25000000\sqrt{110}-1\text{ and }V=0\right)\text{ or }\left(r=25000000\sqrt{110}-1\text{ and }V=0\right)\text{ or }\left(N=0\text{ and }V\neq 0\text{ and }r=25000000\sqrt{110}-1\right)\text{ or }\left(N=0\text{ and }V\neq 0\text{ and }r=-25000000\sqrt{110}-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(r+1\right)^{4}, the least common multiple of 1+r,\left(1+r\right)^{2},\left(1+r\right)^{3},\left(1+r\right)^{4}.

$$NPV\left(r+1\right)^{4}=550000\left(r+1\right)^{3}\times 500000\times \frac{500000}{\left(1+r\right)^{2}}\times \frac{500000}{\left(1+r\right)^{3}}-1000000$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(r+1\right)^{3}.

$$NPV\left(r+1\right)^{4}=550000\left(r^{3}+3r^{2}+3r+1\right)\times 500000\times \frac{500000}{\left(1+r\right)^{2}}\times \frac{500000}{\left(1+r\right)^{3}}-1000000$$

Multiply 550000 and 500000 to get 275000000000.

$$NPV\left(r+1\right)^{4}=275000000000\left(r^{3}+3r^{2}+3r+1\right)\times \frac{500000}{\left(1+r\right)^{2}}\times \frac{500000}{\left(1+r\right)^{3}}-1000000$$

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

$$NPV\left(r+1\right)^{4}=275000000000\left(r^{3}+3r^{2}+3r+1\right)\times \frac{500000}{1+2r+r^{2}}\times \frac{500000}{\left(1+r\right)^{3}}-1000000$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(1+r\right)^{3}.

$$NPV\left(r+1\right)^{4}=275000000000\left(r^{3}+3r^{2}+3r+1\right)\times \frac{500000}{1+2r+r^{2}}\times \frac{500000}{1+3r+3r^{2}+r^{3}}-1000000$$

Express 275000000000\times \frac{500000}{1+2r+r^{2}} as a single fraction.

$$NPV\left(r+1\right)^{4}=\frac{275000000000\times 500000}{1+2r+r^{2}}\left(r^{3}+3r^{2}+3r+1\right)\times \frac{500000}{1+3r+3r^{2}+r^{3}}-1000000$$

Multiply \frac{275000000000\times 500000}{1+2r+r^{2}} times \frac{500000}{1+3r+3r^{2}+r^{3}} by multiplying numerator times numerator and denominator times denominator.

$$NPV\left(r+1\right)^{4}=\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}\left(r^{3}+3r^{2}+3r+1\right)-1000000$$

Use the distributive property to multiply \frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} by r^{3}+3r^{2}+3r+1.

$$NPV\left(r+1\right)^{4}=\frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{3}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Calculate 500000 to the power of 2 and get 250000000000.

$$NPV\left(r+1\right)^{4}=\frac{275000000000\times 250000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{3}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Multiply 275000000000 and 250000000000 to get 68750000000000000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{3}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Express \frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{3} as a single fraction.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Calculate 500000 to the power of 2 and get 250000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+3\times \frac{275000000000\times 250000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Multiply 275000000000 and 250000000000 to get 68750000000000000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+3\times \frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Express 3\times \frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} as a single fraction.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Express \frac{3\times 68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r^{2} as a single fraction.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+3\times \frac{275000000000\times 500000^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Calculate 500000 to the power of 2 and get 250000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+3\times \frac{275000000000\times 250000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Multiply 275000000000 and 250000000000 to get 68750000000000000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+3\times \frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Express 3\times \frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} as a single fraction.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Express \frac{3\times 68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}r as a single fraction.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{275000000000\times 500000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Multiply 275000000000 and 500000 to get 137500000000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{137500000000000000\times 500000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Multiply 137500000000000000 and 500000 to get 68750000000000000000000.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Since \frac{68750000000000000000000r^{3}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} and \frac{3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} have the same denominator, add them by adding their numerators.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}+3\times 68750000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Do the multiplications in 68750000000000000000000r^{3}+3\times 68750000000000000000000r^{2}.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Since \frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} and \frac{3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} have the same denominator, add them by adding their numerators.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}+3\times 68750000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Do the multiplications in 68750000000000000000000r^{3}+206250000000000000000000r^{2}+3\times 68750000000000000000000r.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}+206250000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}+\frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Since \frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}+206250000000000000000000r}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} and \frac{68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)} have the same denominator, add them by adding their numerators.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}+206250000000000000000000r+68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}-1000000$$

Factor the expressions that are not already factored in \frac{68750000000000000000000r^{3}+206250000000000000000000r^{2}+206250000000000000000000r+68750000000000000000000}{\left(1+2r+r^{2}\right)\left(1+3r+3r^{2}+r^{3}\right)}.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000\left(r+1\right)^{3}}{\left(r+1\right)^{2}\left(r+1\right)^{3}}-1000000$$

Cancel out \left(r+1\right)\left(r+1\right)^{2} in both numerator and denominator.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000}{\left(r+1\right)^{2}}-1000000$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1000000 times \frac{\left(r+1\right)^{2}}{\left(r+1\right)^{2}}.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000}{\left(r+1\right)^{2}}-\frac{1000000\left(r+1\right)^{2}}{\left(r+1\right)^{2}}$$

Since \frac{68750000000000000000000}{\left(r+1\right)^{2}} and \frac{1000000\left(r+1\right)^{2}}{\left(r+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000-1000000\left(r+1\right)^{2}}{\left(r+1\right)^{2}}$$

Do the multiplications in 68750000000000000000000-1000000\left(r+1\right)^{2}.

$$NPV\left(r+1\right)^{4}=\frac{68750000000000000000000-1000000r^{2}-2000000r-1000000}{\left(r+1\right)^{2}}$$

Combine like terms in 68750000000000000000000-1000000r^{2}-2000000r-1000000.

$$NPV\left(r+1\right)^{4}=\frac{68749999999999999000000-1000000r^{2}-2000000r}{\left(r+1\right)^{2}}$$

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

$$NPV\left(r+1\right)^{4}=\frac{68749999999999999000000-1000000r^{2}-2000000r}{r^{2}+2r+1}$$

Multiply both sides of the equation by \left(r+1\right)^{2}.

$$NPV\left(r+1\right)^{4}\left(r+1\right)^{2}=68749999999999999000000-1000000r^{2}-2000000r$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$NPV\left(r+1\right)^{6}=68749999999999999000000-1000000r^{2}-2000000r$$

The equation is in standard form.

$$NV\left(r+1\right)^{6}P=68749999999999999000000-2000000r-1000000r^{2}$$

Divide both sides by NV\left(r+1\right)^{6}.

$$\frac{NV\left(r+1\right)^{6}P}{NV\left(r+1\right)^{6}}=\frac{68749999999999999000000-2000000r-1000000r^{2}}{NV\left(r+1\right)^{6}}$$

Dividing by NV\left(r+1\right)^{6} undoes the multiplication by NV\left(r+1\right)^{6}.

$$P=\frac{68749999999999999000000-2000000r-1000000r^{2}}{NV\left(r+1\right)^{6}}$$

Divide 68749999999999999000000-1000000r^{2}-2000000r by NV\left(r+1\right)^{6}.

$$P=\frac{1000000\left(68749999999999999-2r-r^{2}\right)}{NV\left(r+1\right)^{6}}$$