Solving Quadratics the Hard Way

A post to a math Google Group asked about the effect of computerized equation solvers on math education. Don "The Mathman" Cohen replied that during his work with his students he's solved quadratic equations (like x² - 5x + 6 = 0) no less than 12 ways! This list includes continued fractions and iterating using a calculator. Schools normally only teach 3 ways: factoring, completing the square and (my favorite) the quadratic formula.

Being in my calculus mode recently, I wanted to add Newton's method to the list. In the 1600s Newton used his new invention, calculus, to approximate roots of polynomials like the one below:

A "root" of a function is the x-value where the line or curve crosses the x-axis. In the figure above, the root is the red dot. You guess the root is at x₀, then you plug x₀, the function of x₀ and the derivative of the function at x₀ into the formula

x₁ = x₀ - f(x₀)/f'(x₀)

to get x₁, the next approximation for the root. Do that again and x2 is even closer to the red dot in the picture.

I just wrote a short program in Sage to print 9 approximations and used x = 5 as my first guess:


f(x)= x^2-5*x+6
g(x)= derivative (f,x)
a=5
d=1
while d < 10:
    b=a-(f(a)/g(a))
    print N(b)
    a = b
    d = d + 1

Here are the approximations:

3.80000000000000
3.24615384615385
3.04060269627280
3.00152476019449
3.00000231782539
3.00000000000537
3.00000000000000
3.00000000000000
3.00000000000000

Sure enough, 3 is a solution of the equation: 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0. An initial guess of 0 gives the other root of 2. The method gives the root closer to your initial guess.

Newton and his derivatives show up in the Math Through Technology Program starting September 6!