A question about curve intersections --------------------------------------
The thirteenth problem concerns solving algebraic equations and, more specifically, determining their roots using techniques of nomography, that is, as intersections of networks of curves. This is possible if the roots can be expressed as compositions of continuous functions of at most two variables.
Consider any quadratic equation, *ax 2 + bx + c = 0, with a ≠ 0. Its solutions, given by −b±b2−4ac2a,\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}, can be expressed in this way.
If we define f(x,y)=x2y,  g(x,y)=−x+x2−4yf(x, y) = \dfrac{x}{2y}, \; g(x, y) = -x + \sqrt{x^2 -4y} and h (x, y) = xy, then one root can be written as f ( g (b, h ( a, c)), a*); the other can be written similarly.
Using the formulas for the solutions of a cubic equation (see the sidebar "The Cubic Equation, Italian Style" in the article "Before Abel and Galois"), one can see that the method also works for degree 3: the solutions can be expressed as compositions of functions of at most two variables chosen from among the equation's coefficients. In fact, as the German mathematician Ehrenfried Walther von Tschirnhaus (1651–1708) showed, this result holds for equations of degree at most 6.