To solve for the IRR (Internal Rate of Return) in the given Net Present Value equation:

$NPV = -6,750 + \frac{4,500}{(1+IRR)^1} + \frac{18,000}{(1+IRR)^2} = 0$

we can break this down step by step:

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Step 1: Rearrange the equation

$-6,750 + \frac{4,500}{1+IRR} + \frac{18,000}{(1+IRR)^2} = 0$

$\frac{4,500}{1+IRR} + \frac{18,000}{(1+IRR)^2} = 6,750$

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Step 2: Simplify

Let $x = 1 + IRR$. Substitute $x$ into the equation: $\frac{4,500}{x} + \frac{18,000}{x^2} = 6,750$

Multiply through by $x^2$ to eliminate the denominators: $4,500x + 18,000 = 6,750x^2$

Rearrange into standard quadratic form: $6,750x^2 - 4,500x - 18,000 = 0$

Simplify by dividing through by 750: $9x^2 - 6x - 24 = 0$

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Step 3: Solve the quadratic equation

$9x^2 - 6x - 24 = 0$

Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 9$, $b = -6$, $c = -24$: $x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(9)(-24)}}{2(9)}$

Simplify: $x = \frac{6 \pm \sqrt{36 - (-864)}}{18}$

$x = \frac{6 \pm \sqrt{900}}{18}$

$x = \frac{6 \pm 30}{18}$

Two possible solutions: $x = \frac{6 + 30}{18} = \frac{36}{18} = 2$ $x = \frac{6 - 30}{18} = \frac{-24}{18} = -\frac{4}{3}$

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Step 4: Find IRR

Since $x = 1 + IRR$, and $x = 2$ is the only valid solution (as IRR must be positive): $1 + IRR = 2 \quad \implies \quad IRR = 1$

Thus, the IRR = 1 or 100%.

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Let me know if you'd like more clarification or steps explained!

Related Questions:

1. What does IRR represent in financial analysis, and how is it interpreted?
2. How can you use Excel or financial calculators to solve for IRR?
3. What are the limitations of using IRR in project evaluation?
4. How does changing cash flows impact the IRR calculation?
5. What happens if there are multiple IRRs or no IRR?

Tip:

When solving for IRR manually, always check the feasibility of solutions and ensure the results align with the project's financial logic.