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1. **Finding Relationships Between Roots:** Often, problems specify a relationship between the roots (e.g., one root is twice the other, or the roots differ by a certain value). Using Vieta's Formulas, you can set up a system of equations to solve for the roots. For instance, if you have a quadratic equation with roots r₁ and r₂, and you know that r₁ = 2r₂, you can substitute this relationship into the sum and product formulas to find the roots. For example, if you know that the sum is 9 and r₁ = 2r₂, you can set up the equation 2r₂ + r₂ = 9, so 3r₂ = 9, thus r₂ = 3. Therefore, r₁ = 6. This can then be used to find the coefficients of the original equation.
