We start by trying to solve a system of two equations, but plugging in whole and real numbers reveals a dilemma: the answers always seem to sway far apart. Negative numbers do not help either, since no two negative numbers multiply to a positive result that can offset the difference.
So we turn to algebra. We rearrange the first equation to solve for y, substitute it into the second, expand, and rewrite the result in standard quadratic form. The resulting graph never crosses zero on the x-axis, which suggests the equation has no real solution. Applying the quadratic formula confirms the problem: the square root of a negative number (-108) appears, which the real number system cannot handle. We have hit a wall.
This same wall stopped the Italian mathematician Gerolamo Cardano in 1545. While developing general solutions to cubic equations, he found that even when an equation yielded a real, tangible answer, the algebraic path forced him to calculate the square root of a negative number. In Ars Magna, published that same year, he asked readers to divide 10 into two parts whose product was 40.
Applying the quadratic formula to that problem produced a root of -15, which seemed impossible to nearly every mathematician of his time. Cardano discovered that if he stubbornly followed the standard rules of algebra with these forbidden quantities, the imaginary parts would eventually vanish, leaving a perfectly accurate real solution. That discovery opened the door to complex numbers.
Source: Hacker News · Summarized by HeadlinesBriefing