Studying Mathematics Thread

[GrandizerII]

Speaking of proofs, since I am an amateur compared to you guys, I'll do something relatively basic. Let's see how to come up with the quadratic formula (I am doing this now without checking any sources, and it's been more than a year since I last did some serious algebra studying/practices).

ax^2 + bx + c = 0 (starting equation from which to eventually derive the quadratic formula, with coefficient 'a' being non-zero)

x^2 + b/a(x) + c/a = 0 (divide both sides of the equation by 'a')

x^2 + b/a(x) + (b/(2a))^2 - (b/(2a))^2 + c/a = 0 (complete the square and add in the compensatory term to keep the equation true)

x^2 + b/a(x) + (b/(2a))^2 = (b/(2a))^2 - c/a (move all terms not containing 'x' to the right-hand side of the equation)

(x + (b/(2a)))^2 = b^2/(4a^2) - c/a (factor the left-hand side and expand the first term of the right-hand side)

x + b/(2a) = +- sqrt(b^2/(4a^2) - c/a) (reverse the square on the left-hand side in the proper manner)

x + b/(2a) = +- sqrt((b^2-4ac)/(4a^2)) (unify the fractions within the square root)

x + b/(2a) = +- sqrt(b^2-4ac)/(2a) (pull the denominator of the right-hand side fraction out of the square root)

x = -b/(2a) +- sqrt(b^2-4ac)/(2a) (move the second term of left-hand side to the right-hand side)

x = (-b +- sqrt(b^2-4ac))/(2a) (combine the two fractions on the right-hand side, and we finally have the infamous quadratic formula)