The quadratic formula is used to find roots of second-degree ("quadratic") polynomials, i.e. expressions of the form a * x^2 + b * x + c .

For the above polynomials, the roots are: x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

Proof (completing the square method):

ax² +bx + c = 0
ax² + bx = -c
x² + bx / a = -c/a
x² + bx / a + (b / 2a)² = -c / a + (b / 2a)²
(x + (b / 2a))² = -4ac / 4a² + b² / 4a²
x + (b / 2a)² = (b² - 4ac) / 4a²
x + (b / 2a) = +/- sqrt ((b² - 4ac) / 4a²)
x = -b / 2a +/- sqrt(b² - 4ac) / 2a
x = -b +/- sqrt(b² - 4ac) / 2a

In my Algebra 2 class, our crazy teacher taught us a song where you sing the quadratic formula to the tune of "Pop Goes The Weasel".

X equals the opposite of B
Plus or minus the square root
of b squared minus 4 A C
All over 2A!

I can't tell you why it worked so well, but to this day, I can never forget it...

Most commonly written as something somewhat resembling:

           .----------
 -b +  _  /  b2 - 4ac
        \/             
----------------------
          2a

My stupidity now corrected thanks to JustSomeGuy

If ax² + bx + c = 0, then:

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

Want to use this on your fancy TI-83? Use this program:

:a+bi
:Lbl 1
:ClrHome
:Input "A=",A
:Input "B=",B
:Input "C=",C
:If A=0:Goto 1
:(-B+root(B2-4AC))/(2A) -> X
:(-B-root(B2-4AC))/(2A) -> Y
:Disp "ANSWERS STORED"
:Disp "AS X AND Y"
:Disp X
:Disp Y

Where -> indicates the "store" operator, a+bi indicates the command that sets a+bi mode and root() indicates the square root operator.

The proof of the quadratic equation is delightfully simple to learn. It's also the only proof I had to memorise for College Algebra:

                   Start with: ax2 + bx + c = 0

                   Subtract c: ax2 + bx = -c

                  Divide by a: x2 + bx/a = -c/a

          Complete the Square: x2 + bx/a + b2/4a2 = -c/a + b2/4a2

   Factor left, combine right: (x+b/2a)2 = (b2-4ac)/4a2

                  SQRT it all: x+b/2a = ±SQRT(b2-4ac)/2a

Subtract b/2a from both sides: x = [-b ±SQRT(b2-4ac)]/2a

Sometimes called the quadratic formula. (Probably a better name since it's used to solve quadratic equations.)

The numerical properties of the Quadratic Equation make it unsuitable for general use in software. For example, consider the special case a=0, which a human would obviously recognize. For a computer, special code must be written. More subtly, from Numerical Analysis, we must consider the possibility of cancellation error when b² is much greater than 4ac. In this case, the solution for one root will be very close to zero, and the other will be close to -b/a. The near-zero number will suffer accuracy problems.

To avoid the difficulty with cancellation, we may compute the roots with the following:

z = -0.5 (b + sign(b)*(b² - 4ac))

x₁ = z/a
x₂ = c/z

Source: Numerical Recipes

In my Algebra 2 class, we learned the quadratic formula to the tune of "Gillligan's Island":
Negative B plus or minus
The square root of
B-squared minus 4ac
All over 2a
All over 2a

Other people claimed to have learned the handy formula by singing it to the tune of:
Frère Jacques
Pop Goes the Weasel
Row, Row, Row Your Boat
God Rest Ye Merry Gentlemen
Notre Dame Fight Song
U of M Fight Song
Jingle Bells
Ballin' the Jack
Amazing Grace