Method 2: The root formula of the unary cubic equation that can't be factorized can't be worked out by ordinary deductive thinking. The standard unary cubic equation of AX 3+BX 2+CX+D = 0 can only be formalized as a special type of X 3+PX+Q = 0 by matching method similar to the root formula of the unary quadratic equation.
Caldan formula
Univariate cubic equation x 3+px+q = 0 (p, q∈R) Discriminant δ = (q/2) 2+(p/3) 3 Caldan formula x1= (y1) (1) X3 = (y1) (1/3) ω 2+(y2) (1/3) ω, where ω = (-1+i3 (1/2))/2; Y( 1，2)=-(q/2)((q/2)^2+(p/3)^3)^( 1/2)。 ) The general univariate cubic equation AX 3+BX 2+CX+D = 0 makes X = Y-B/(3A) substituted into the above equation, which can be transformed into a special cubic equation Y 3+Py+Q = 0 suitable for solving Kardan's formula.
Jinsheng formula
Cubic equation is widely used. Solving a univariate cubic equation with root sign, although there is a famous Caldan formula and the corresponding discrimination method, it is more complicated and lacks intuition to solve the problem with Caldan formula. Shenjin Fan derived a set of simple root-seeking formulas of cubic equation with one variable directly expressed by A, B, C and D, and established a new discriminant method. The unary cubic equation ax 3+bx 2+CX+d = 0, (a, b, c, d∈R, a≠0). Multiple root discriminant: a = B2-3ac；; B = 9adC = c 2-3bd BC, and the total discriminant is δ = b 2-4ac. When A=B=0, the golden formula ①: x (1) = x (2) = x (3) =-b/(3a) =-c/b =-3d/c. When δ = b 2-4ac >; 0, the formula of gold ②: x (1) = (-b-y (1) (1/3)-y (2) (1/3))/(3a); x(2,3)=(-2b+y⑴^( 1/3)+y⑵^( 1/3))/(6a)i3^( 1/2)(y⑴^( 1/3)-y⑵^( 1/3))/(6a)； Where y (1, 2) = ab+3a (-b (B2-4ac) (1/2))/2, I 2 =-1. When δ = b 2-4ac = 0, formula ③: x (1) =-b/a+k; X = x3 =-k/2, where K=B/A, (A≠0). When δ = b 2-4ac; 0，- 1 & lt； T< 1) gold discrimination method: when A=B=0, the equation has triple real roots; ②: when δ = b 2-4ac >; 0, the equation has a real root and a pair of * * * yoke imaginary roots; ③: When δ = b 2-4ac = 0, the equation has three real roots, including one multiple root; ④ When δ = b 2-4ac 0 (at this time, apply Jinsheng formula ② to solve the problem). Golden Sage Theorem 5: When a < 0, there must be δ > 0 (at this time, apply Golden Sage Formula ② to solve the problem). Jinsheng Theorem 6: When δ = 0, if B=0, there must be A=0 (at this time, apply Jinsheng formula ① to solve the problem). Jinsheng Theorem 7: When δ = 0, if B≠0, Jinsheng Formula ③ must have no value of A≤0 (in this case, Jinsheng Formula ③ should be used to solve the problem). Jinsheng Theorem 8: When δ < 0, Jinsheng Formula ④ must have no value of A≤0. (At this time, apply Jinsheng Formula ④ to solve the problem). Jinsheng Theorem 9: When δ < 0, T≤- 1 or T≥ 1 in Jinsheng Formula ④ must have no value, that is, the value of T must be-1 < t < 1. Obviously, when A≤0, there is a corresponding formula to solve the problem. Note: The inverse of the Golden Sage Theorem may not be true. For example, when δ > 0, there is not necessarily < 0. The Golden Sage Theorem shows that the Golden Sage formula is always meaningful. Jin Sheng formula can be used to directly solve the univariate cubic equation with arbitrary real coefficients. When δ= 0(d≠0), when δ= 0, Jinsheng formula has no radical sign, and the efficiency of solving the equation is still high. Compared with Kadan formula, Jinsheng formula is simpler to express, and it is more intuitive and efficient to use Jinsheng formula to solve problems. Using the discrimination method of gold wealth, the solution of the discrimination equation is intuitive. Multiple root discriminant A = B2-3ac；; B = 9adC = c 2-3bd is the simplest formula, and the total discriminant δ = b 2-4ac composed of A, B and C is also the simplest formula (it is a very beautiful formula), and its shape is the same as the discriminant of the root of a quadratic equation. The formula (-b (b 2-4ac) (1/2))/2 in jinsheng formula ② has the form of finding the root of a quadratic equation, and these expressions reflect the order, symmetry, harmony and conciseness of mathematics.