A quadratic equation is of the form ax^2 + bx + c = 0. By the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n roots (counting multiplicity).
By the Fundamental Theorem of Algebra, the number of roots of a polynomial equation is equal to its highest degree. Since a quadratic equation is defined as a polynomial equation of degree two, it can have a maximum of two solutions. Therefore, the maximum number of solutions is two.