Final answer:
Completing the square and using the quadratic formula are the most reliable methods for solving quadratic equations, applicable to any quadratic and providing precise solutions compared to factoring, graphing, or guessing.
Step-by-step explanation:
When solving quadratic equations, reliability is key in selecting methods. Two methods stand out for their universal applicability and mathematical soundness: completing the square and using the quadratic formula. These methods work for any quadratic equation, unlike factoring which requires the equation to be factorable, and graphing which provides approximate solutions.
Completing the square transforms a quadratic equation into a perfect square trinomial, allowing for the direct extraction of the variable's value. The quadratic formula, derived from completing the square, provides the roots directly from the coefficients of the equation and is particularly useful when the solutions are not integers or simple fractions. Factoring is quick and efficient when applicable; however, it is limited to equations that can be factored easily. Graphing is insightful for visualizing the parabolic nature of quadratics but is less precise for finding exact solutions.
Guessing and checking, substitution, and elimination are methods that can sometimes be employed for systems of equations, with guessing being the least precise and practical. Substitution and elimination are more suited to linear systems than quadratic ones. Therefore, for solving quadratic equations, completing the square and the quadratic formula are the most reliable methods.