Envelopes are Solving Machines for Quadratics and Cubics and Certain Polynomials of Arbitrary Degree

Everybody knows from school how to solve a quadratic equation of the form $x^2-px+q=0$ graphically. But this method can become tedious if several equations ought be solved, as for each pair $(p,q)$ new parabola has drawn. Stunningly, there is one single curve that used every via drawing tangent lines through given point curve. In article we derive in an elementary way and generalize it $x^n-px+q=0$ arbitrary $n \ge 2$. Moreover, number solutions specific seen immediately with technique. Concluding out connections duality points plane concept Legendre transformation.