Newton's Method — Chase a Root Down Tangent Lines You Draw

How do you solve an equation with no tidy formula? Around 1669 Isaac Newton gave an answer: guess, draw the tangent line to the curve there, and follow it down to where it crosses zero -- that landing point is a much better guess. Repeat, and you zoom in. You will do it with a ruler.

Rule a grid on card (your graph paper). Plot the curve y = x-cubed minus 2x minus 5 for x from 1.5 to 3 by working out a few points and joining them smoothly. It crosses the x-axis somewhere near x = 2 -- that crossing is the solution you are hunting.

Start at x = 2. Lay your ruler along the curve there to draw the tangent line, and mark where that straight line crosses the x-axis -- read off the new x. Move to that x on the curve, draw the new tangent, and mark where IT crosses. After just two or three tangents your marks pile up on the root, near x = 2.095.

What your tangents teach. (1) Each step replaces the guess with x minus f(x) divided by the slope f'(x); near the root the accuracy doubles every step, dazzlingly fast. (2) Heron's ancient square-root trick is just Newton's method applied to 'x squared minus S'. (3) It needs a derivative (the slope) and a reasonable starting guess -- start in a bad spot, or near a flat part of the curve, and the tangents can fly AWAY from the root instead of toward it. (4) Given a good start, it is the default way computers solve equations in engineering, physics, computer graphics and the training of machine-learning models.