• ax^2 + bx + c = 0
• ax^2 + bx = -c move the c over
• x^2 + (b/a)x = -c/a divide by a
• x^2 + (b/a)x +(b/2a)^2 = -c/a + (b/2a)^2 complete the square
• (x + b/2a)^2 = -c/a + (b/2a)^2 factor the left hand side
• x + b/2a = sqrt(-c/a + (b/2a)^2) now we just tidy it up
• x = -b/2a + sqrt(-c/a + b^2/4a^2)
• x = -b/2a + (2a/2a) sqrt(-c/a + b^2/4a^2)
• x = (-b + (2a)sqrt(-c/a + b^2/4a^2))/2a
• x = (-b + sqrt(-4ac + b^2))/2a move 2a into the square root and multiply it with what's inside

The derivation of the quadratic formula is nice because it doesn't rely on anything fancy and it's all tricks the teacher is likely to teach around the same time you're learning it. It's not voodoo shit, it's just the ax^2 + bx + c = 0 and you solve for x.