The mathematician Girolamo Cardano is credited with the first use (in 1545) of negative sq...

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 5 + i115 and 5 − i115 satisfy the conditions of the and whose product is 40.” Show that the complex numbers  and  satisfy the conditions of the problem. (Cardano did not use the symbolism  or even . He wrote R.m 15 for , 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.”)