There are four basic methods for solving trinomial quadratic equations: graphing, factoring, quadratic formula, and completing the square. This post gives an example of how to complete the square for the three basic species of quadratic equation that it is applied to. The first type has a leading coefficient of 1. The second type has a leading coefficient other than one, which can be reduced to 1 in the second step. The third type has a leading coefficient other than 1, which cannot be reduced in the second step. It takes practice to learn how to complete the square, which is really a back door alternative to using the quadratic formula.
Type I: Leading Coefficient = 1
Type II: Leading Coefficient Other Than 1, Eliminated in Second Step
Type III: Leading Coefficient Other Than 1, Eliminated After Second Step
Since 2008, when I started my private practice as a math tutor in Las Vegas, I have used a unified formula that captures all six of the possible transformations that can be applied to a function. Textbooks, by contrast, typically teach the transformations separately. The photo above was well received by a calculus student who utilized my math tutoring services for a refresher in precalculus. From left to right, the possible transformations are as follows:
Mr. Ken Marciel