I understand the quadratic equation, axis of symmetry, vertex, and intercepts, but how does that all work? Why -b/2a and not -c? Why over 2a, not 2c?

Answer 1

try completing the square and deriving the quadratic formula yourself

that way if you ever forget it or aren't sure you know it correctly,

you can derive it yourself, within less than a couple minutes:

ax^2 +bx + c = 0

a(x^2 +bx/a) = -c

a(x^2 +bx/a + (b/2a)^2) = - c + b^2/4a

a(x-b/2a)^2 = b^2/4a -c

(x+b/2a)^2 = b^2/4a^2 - c/a

(x+b/2a)^2 = b^2/4a^2 - 4ac/4a^2

(x+b/2a)^2 = (b^2 -4ac)/4a^2

x +b/2a =+/- sqr(b^2-4ac)/2a

x = -b/2a + or - sqr(b^2-4ac)/2a

x= (-b +/- sqr(b^2 -4ac))/2a