The mathematician Girolamo Cardano is credited with the first use (in 1545) of negative square roots in solving the now-famous problem, “Find two numbers whose sum is 10 and whose product is 40.” Show that the complex numbers

satisfy the conditions of the problem. (Cardano did not use the symbolism i√15 or even √ - 15. He wrote R.m 15 for √ - 15 meaning “radix minus 15.” He regarded the numbers 5 + R.m 15 and 5 - R.m 15 as “fictitious” or “ghost numbers,” and considered the problem “manifestly impossible.” But in a mathematically adventurous spirit, he exclaimed, “Nevertheless, we will operate.”)

Transcribed Image Text:

5 + i√15 and 5 - iV15