start with quadratic equation in the form of ax^2 + bx + c = 0
divide both sides by a --> x^2 + (b/a)x + (c/a) = 0
bring c/a to the right --> x^2 + (b/a)x = (-c/a)
take 1/2 of b/a --> 1/2 * b/a = b/2a
square b/2a --> b^2/(4a^2)
the b^2/(4a^2) is used to complete the square
add b^2/(4a^2) to both sides --> x^2 + (b/a)x + b^2/(4a^2) = (-c/a) + b^2/(4a^2)
note by FOIL (first outer inner last) -->
(x + b/(2a))^2 = x^2 + (b/a)x + b^2/(4a^2)
simplify x^2 + (b/a)x + b^2/(4a^2) = (-c/a) + b^2/(4a^2)
(x + b/(2a))^2 = (-c * 4a)/(a * 4a) + b^2/(4a^2)
(x + b/(2a))^2 = (b^2 - 4ac)/(4a^2)
take square root of both sides
x + b/(2a) = +/- sqrt(b^2 - 4ac)/(2a)
move b/(2a) to the right
x = (-b +/- sqrt(b^2 - 4ac))/(2a), this is also the quadratic formula
(