1. **State the problem:** Solve the quadratic equation $$q_i^2 + 2q_i p_i - 2q_i = 0$$ for $q_i$, where $q_i$ and $p_i$ are allele frequencies. 2. **Rewrite the equation:** Factor out $q_i$ from all terms: $$q_i^2 + 2q_i p_i - 2q_i = q_i(q_i + 2p_i - 2) = 0$$ 3. **Apply the zero product property:** For the product to be zero, either $$q_i = 0$$ or $$q_i + 2p_i - 2 = 0$$ 4. **Solve the linear equation:** $$q_i + 2p_i - 2 = 0$$ Subtract $2p_i$ from both sides: $$q_i + \cancel{2p_i} - 2 - \cancel{2p_i} = 0 - 2p_i$$ which simplifies to $$q_i - 2 = -2p_i$$ Add 2 to both sides: $$q_i - 2 + 2 = -2p_i + 2$$ which simplifies to $$q_i = 2 - 2p_i$$ 5. **Final solutions:** $$q_i = 0 \quad \text{or} \quad q_i = 2 - 2p_i$$ 6. **Interpretation:** These are the two possible values for the frequency $q_i$ given the frequency $p_i$ of other alleles. --- **Summary:** The quadratic equation factors to give two solutions for $q_i$: zero or $2 - 2p_i$.