Chapter 2.1, Problem 63PE

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 $5+i\sqrt{15}\text{ and }5-i\sqrt{15}$ 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.”)