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he used algebra & geometry
The Quadratic formula in mathematics is used to solve quadratic equations in algebra. The simplest way to solve these equations is to set each of the factors to zero and then solve each factor separately.
I suspect that different people - or schools - may use the term "intermediate mathematics" for different things.
It is Arithmetica, by Diophantus of Alexandria, who is generally considered the father of algebra, that may be the first writing on algebra. But there is a bit of debate over who the father of algebra is. Use the link below to the related question.
As an automotive engineer, I mostly use algebra, but I sometimes use geometry, statistics, and calculus. Some higher level research positions may use differential equations.