Chapter 1.4, Problem 77E

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 5 + i $\sqrt{15}$ and 5 − i $\sqrt{15}$ satisfy the conditions of the problem. (Cardano did not use the symbolism i $\sqrt{15}$ or even $\sqrt{-15}$. He wrote R.m 15 for $\sqrt{-15}$, 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.”)