DERIVING THE QUADRATIC FORMULA. Trivial.

ax2 + bx + c = 0
Divide both sides by a.

x2 + (b/a)x + c/a = 0
Subtract c/a from both sides.

x2 + (b/a)x = -c/a
Divide the coefficient of the x term, b/a, by two and then square it. Add this to both sides.

(b/2a)2

b2/4a2

x2+(b/a)x+b2/4a2 = -c/a + b2/4a2
Simplify by factoring the left side and adding terms on the right side (find a common denominator first).

(x + b/2a)(x + b/2a) = (-4ac/4a2) + (b2/4a2)

(x + b/2a)2 = (b2 - 4ac)/4a2
Take the square root of each side.

√((x + b/2a)2) = ±√((b2 - 4ac)/4a2)

x + b/2a = ±√(b2 - 4ac)/2a
Subtract b/2a from both sides and you will at last have the quadratic formula.

x = (-b ±√(b2 - 4ac))/2a