graph TD
    A[Quadratic equation problem] --> B{What is asked?}
    B -->|Nature of roots| C[Compute D = $b^{2-4ac}$ and analyze sign]
    B -->|Find roots| D[Use quadratic formula or factorization]
    B -->|Root expressions| E[Use Vieta's: sum = -b/a, product = $\frac{c}{a}$]
    B -->|Location of roots| F[Apply conditions: D, f at points, vertex]
    B -->|Form new equation| G[Find new sum and product, or use substitution]
    B -->|Inequality| H[Find roots and use sign chart]
    B -->|Common root| I[Subtract equations or use condition formula]
    E --> J[$alpha^{2+beta}^2$ = $sum^2$ - 2*product]
    E --> K[1/alpha + 1/beta = $\frac{sum}{product}$]
    F --> L[f of k < 0 means k between roots]