Comparing Methods for Solving Quadratics

Key Questions

  • Best method to solve quadratic equations.
    There are so far 8 common methods to solve quadratic equations, They are: graphing, completing the squares, quadratic formula, factoring FOIL, The Diagonal Sum Method, the Bluma Method, the popular factoring AC Method, and the new Transforming Method.
    - When the quadratic equation can't be factored, the quadratic formula is the obvious choice. However, solving by formula feels like boring and repeating. In addition, school math curriculum wants students to learn, beyond the formula, a few other solving methods. That is why many quadratic equations given in problems/tests/exams are intentionally set up so that students have to solve them by other solving methods.
    - When the quadratic equations can be factored, the new Transforming Method (Google Search) would be the best choice. It is simple, fast, systematic, no guessing, no factoring by grouping, and no solving the 2 binomials.
    Example: Solve: #y = 16x^2 - 62x + 21 = 0#.
    Solve the transformed equation #y' = x^2 - 62 x + 336 = 0#. Compose factor pairs of (ac = 336) --> ...(4, 82)(6, 56). This last sum is 62 = -b. Then, the 2 real roots of y' are: 6 and 56.
    The 2 real roots of original y are: #x1 = 6/a = 6/16 = 3/8#, and
    #x2 = 56/a = 56/16 = 7/2#