一元三次方程新型解法（A new solution of the three equation of a dollar）.doc

一元三次方程新型解法（A new solution of the three equation of a dollar） Cubic equation solution - Sheng Jin formula solutions Using the formula and Sheng Jin Sheng Jin discriminance and Shengjins theorems introduced from here to you Cubic equation is widely used. Solution of a cubic equation by radicals, although cardans formula famous, and there is a corresponding discriminant approach, but the use of Cartan formula to solve more complex, less intuitive. Van Shengjin derived a set of a, B, C directly with the expression of D and the general formula is a concise form of new unary cubic equation, and established a new discriminant method. Sheng Jin formula Sheng Jin formula A cubic equation AX^3bX^2cXd=0, (a, B, C, D, R, and a = 0). Heavy root discriminant: A=b^2 3aC; B=bc 9ad; C=c^2 3bd, The total discriminant: Delta =B^2 4AC. When A=B=0, Sheng Jin formula: X1=X2=X3=b/ (3a) =c/b=3d/c. When =B^2 4AC0, the formula of Sheng jin: X1= (B (Y1) ^ (1/3) - (Y2) ^ (1/3)) / (3a); X2, X3= (2B (Y1) ^ (1/3) + (Y2) ^ (1/3)) / (6a) + i3^ (1/2) ((Y1) ^ (1/3) - (Y2) ^ (1/3)) / (6a), The Y1, Y2=Ab3a (B + (B^24AC) ^ (1/2)) /2, I^2=1. When =B^2 4AC=0, the formula of Sheng jin: X1=b/aK; X2=X3=K/2, The K=B/A (A = 0). When =B^2 4AC0, the formula of Sheng jin: X1= (B2A^ (1/2) cos (/3 0)) / (3a); X2, X3= (BA^ (1/2) (COS (0 /3) + 3^ (1/2) sin (/3 0))) / (3a), The 0 =arccosT T= (2Ab3aB) / (2A^ (3/2), (A0), 1T1). Sheng Jin discriminant method Sheng Jin discriminant method 1: when A=B=0, the equation has a three shigemi root; The delta =B^2: when 4AC0, the equation has a real root and a pair of complex conjugate roots; The delta =B^2: when 4AC=0, the equation has three real roots, which has a double root; The delta =B^2: when 4AC0, equation three unequal real roots. Sheng Jin theorem Sheng Jin theorem When b=0, c=0, Sheng Jin formula of no significance; when A=0, the formula of Sheng Jin meaningless; when A = 0, the formula of Sheng Jin meaningless; when -1 or T T 1, Sheng Jin formula does not have the s