2. Let us now consider such cubic equations as have all their terms, and which are therefore of this form, x^3 + Ax^2 + Bx + C = 0, where A, B and C denote known quantities, either positive or negative. This equation may be transformed into another in which the second term is wanting by the substitution x = y - A/3. This transformation is a particular case of a general theorem. Let x^n + Ax^(n - 1) + Bx^(n - 2) ... = 0. Substitute x = y + h; then (y + h)^n + A(y + h)^(n - 1) ... = 0. Expand each term by the binomial theorem, and let us fix our attention on the coefficient of y^(n - 1). By this process we obtain 0 = y^n + y^(n - 1)(A + nh) + terms involving lower powers of y. Now h can have any value, and if we choose it so that A + nh = 0, then the second term of our derived equation vanishes. Resuming, therefore, the equation y^3 + qy + r = 0, let us suppose y = v + z; we then have y^3 = v^3 + z^3 + 3vz(v + z) = v^3 + z^3 + 3vzy, and the original equation becomes v^3 + z^3 + (3vz + q)y + r = 0. Now v and z are any two quantities subject to the relation y = v + z, and if we suppose 3vz + q = 0, they are completely determined. This leads to v^3 + z^3 + r = 0 and 3vz + q = 0. Therefore v^3 and z^3 are the roots of the quadratic t^2 + rt - q^2/27 = 0. Therefore v^3 = -1/2 r + [root][(1/27)q^3 + 1/4 r^2]; z^3 = -1/2 r - [root][(1/27)q^3 + 1/4 r^2]; v = [root 3]{-1/2 r + [root][(1/27)q^3 + 1/4 r^2]}; z = [root 3]{-1/2 r - [root][(1/27)q^3 + 1/4 r^2]}; and y = v + z = [root 3]{-1/2 r + [root][(1/27)q^3 + 1/4 r^2]} + [root 3]{-1/2 r - [root][(1/27)q^3 + 1/4 r^2]}. Thus we have obtained a value of the unknown quantity y, in terms of the known quantities q and r; therefore the equation is resolved. 3. But this is only one of three values which y may have. Let us, for the sake of brevity, put A = -1/2 r + [root]((1/27)q^3 + 1/4 r^2), B = -1/2 r - [root]((1/27)q^3 + 1/4 r^2), and put [alpha] = 1/2(-1 + [root]-3), [beta] = 1/2(-1 - [root]-3). Then, from what has been shown (S 1), it is evident that v and z have each these three values, v = [root 3]A, v = [alpha][root 3]A, v = [beta][root 3]A; z = [root 3]B, z = [alpha][root 3]B, z = [beta][root 3]B. To determine the corresponding values of v and z, we must consider that vz = -(1/3)q = [root 3](AB). Now if we observe that [alpha][beta] = 1, it will immediately appear that v + z has these three values, v + z = [root 3]A + [root 3]B, v + z = [alpha][root 3]A + [beta][root 3]B, v + z = [beta][root 3]A + [alpha][root 3]B, which are therefore the three values of y. The first of these formulae is commonly known by the name of Cardan's rule (see ALGEBRA: _History_).