Although the symbols we use here are relatively recent, the mathematics is so old that the original discoverers (or inventors?) are unknown. This kind of work is typical of algebra up to about the start of the 19th century. At that time the ``state of the art'' was roughly as follows:
is not solvable in this fashion.
The work of Abel and Galois, as well as subsequent work of Cayley, Sylvester, Hamilton, Boole, etc. was the start of what is now known as ``modern'' or ``abstract'' algebra. The basic idea of modern algebra is not only to understand how to manipulate expressions, but rather to focus on the underlying algebraic structures that allow you to make those manipulations.
For instance in solving quadratic equations by
the Quadratic Formula, we usually assume that 
are real
numbers. Then the ``usual rules of algebra'' for the addition and
multiplication of real numbers - the commutativity
and associativity of multiplication, the
distributive
law for multiplication over addition, the existence of multiplicative
inverses for nonzero 
, the existence of square roots for
all non-negative real numbers, etc. - are what allow us to derive
the Quadratic Formula. Listing all these properties (but omitting the
existence of square roots!) provides the definition
of the algebraic structure known as a field. We can study
fields ``in the abstract'' and ask:
is one; the existence
of an ordering relation that lets you make the comparison 
is another!)The last question, especially, led to what can only be called a revolution in thinking about what algebra is and what it does.