The ancient Greek mathematicians (2500 years ago) used geometric methods to solve quadrati...

The ancient Greek mathematicians (2500 years ago) used geometric methods to solve quadratic equations or, rather, to construct line segments whose lengths were the roots of the equations. According to historian Howard Eves in An Introduction to the History of Mathematics, 6th ed. (Philadelphia: Saunders College Publishing, 1990):

Imbued with the representation of a number by a length and completely lacking any adequate algebraic notation, the early Greeks devised ingenious geometrical processes for carrying out algebraic operations.

One of the methods used by the ancient Greeks to solve a quadratic equation is described in the following example (using modern algebraic notation).

To construct a line segment whose length is equal to the (positive) root of the equation x2 + 8x = 84:

Begin with a line segment  of length 8. At B, construct a line segment  such that the length of  is . Next, let M be the midpoint of . With M as center, draw a circular arc of radius  intersecting  (extended) at P. Then the length of  is the required root.

On your own or with a group of classmates, work through the preceding construction. That is, sketch the appropriate figure and verify that the construction indeed yields the positive root of the equation. Then, on your own, use the Greek method to determine the positive root of each of the following equations. Write out your work in detail, as if you were explaining it to a student who had not seen

it before. Be sure to include the appropriate geometric figures and an explanation of why the method works.

(a) x2 + 14x = 72

(b) x2 + 2cx = d (c, d > 0)