Square roots of negative numbers came up when solving cubic equations, even if the final solutions were all real. This meant the square root of a negative number was not something nonsensical the way you might claim for x^2 = -1, but actually...real in some sense.

Specifically I believe it involved a geometric construction for solving the cubics, which in some cases could not find a solution unless you allowed a square with "negative area".

There's a good YouTube video on it that includes an epic math battle.

Veritasium - How Imaginary Numbers Were Invented - https://youtu.be/cUzklzVXJwo

Solving the cubic was a physical thing back then. https://www.maa.org/press/periodicals/convergence/solving-th...