1. cardan formula method
(cardano formula method)
Special type of unary cubic equation x 3+px+q = 0 (p, q∈R)
Discriminant δ = (q/2) 2+(p/3) 3
Caldan formula
x 1=(y 1)^( 1/3)+(y2)^( 1/3)；
x2 =(y 1)^( 1/3)ω+(y2)^( 1/3)ω^2； A real root of Caldan formula in standard equation
x3=(y 1)^( 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 standard unary cubic equation ax 3+bx 2+CX+d = 0
Let x = y-b/(3a) be substituted into the above formula,
It can be transformed into a special one-dimensional cubic equation y 3+py+q = 0, which is suitable for directly solving Kardan formula.
Cardin discriminant method
When δ = (q/2) 2+(p/3) 3 >; 0, the equation has a real root and a pair of * * * yoke imaginary roots;
When δ = (q/2) 2+(p/3) 3 = 0, the equation has three real roots, one of which is a multiple root;
When δ = (q/2) 2+(p/3) 3
2. The Golden Sage Formula Method
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.
Jinsheng formula
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,
Total discriminant: δ = b 2-4ac.
When A=B=0, Jin Sheng formula ①:
X⑴=X⑵=X⑶=-b/(3a)=-c/b=-3d/c .
When δ = b 2-4ac >; 0, jinsheng formula ②:
x⑴=(-b-y⑴^( 1/3)-y⑵^( 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= B2-4ac = 0, the golden formula ③:
x⑴=-b/a+K； X⑵=X3=-K/2，
Where K=B/A, (A≠0).
When δ = b 2-4ac
x⑴=(-b-2a^( 1/2)cos(θ/3))/(3a)；
x(2,3)=(-b+a^( 1/2)(cos(θ/3)3^( 1/2)sin(θ/3)))/(3a)；
Where θ=arccosT, t = (2ab-3ab)/(2a (3/2)), (a >；; 0，- 1 & lt； T & lt 1)
Gold filling identification method
①: 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
Jinsheng theorem
When b=0 and c=0, the formula of "Jin Sheng" is meaningless; When A=0, the golden formula ③ is meaningless; When A≤0, formula ④ is meaningless; When t 1, formula ④ is meaningless.
When b=0 and c=0, does formula ① hold? Is there a value of A≤0 in Formula ③ and Formula ④? Is there a value of t 1 in jinsheng formula ④? The Golden Sage Theorem gives the following answers:
Jinsheng Theorem 1: When A=B=0, if B=0, there must be c=d=0 (at this time, the equation has triple real roots of 0, and Jinsheng formula ① still holds).
Golden Sage Theorem 2: When A=B=0, if b≠0, there must be c≠0 (in this case, use Golden Sage Formula ① to solve the problem).
Golden Sage Theorem 3: When A=B=0, there must be C=0 (at this time, apply Golden Sage Formula ① to solve the problem).
Jinsheng Theorem 4: When A=0, if B≠0, there must be δ > 0 (in this case, Jinsheng Formula ② is used 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), there is still a prescription to solve the problem by using Caldan formula. 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.