Question

Construction qudirertiy

Answer 1

A quadratic equation

ax² + bx + c = 0,

with the leading coefficient a ≠ 0, has two roots that may be real - equal or different - or complex. The roots can be found from the quadratic formula:

x1,2 = (-b ~+mn~ √b² - 4ac) / 2a,

(On a more extended discussion of solving and graphing the quadratic equation see the article Graph and Roots of Quadratic Polynomial.)

In addition to the four arithmetic operations, the formula includes a square root. This is exactly the kind of operations that produce constructible numbers from constructible numbers. (By being "constructible" we mean a length (of a straight line) segment that could be constructed by straightedge and compass, given a unit length.) In fact, Descartes used this example to introduce his ideas of analytic geometry.