Is the Sheng Jin formula feasible? An example of how to solve Shengjin s formula

Is it feasible to solve a cubic equation using Sheng Jin's formula? Is there a publication in a journal?

The solution of the Shengjin formula of the unary cubic equation was published in the Journal of Hainan Normal University (Natural Science Edition) (Vol. 2, No. 2; Hainan, China, December 1989), pp. 91-98. Author: Fan Shengjin.

See the derivation of Shengjin's formula and the establishment of Shengjin's discriminant method.

It's almost solvable, but it doesn't feel reliable enough.

Steps to solve the problem using Shengjin's formula:

1. Write the values of the coefficients a, b, c, and d (so as not to mistakenly input the value of c into the calculator as the value of b when b=0);

2. Find the values of a, b, c, and δ in order;

3. According to the Shengjin discriminant method, the correct result can be obtained by applying the corresponding Shengjin formula.

Example: (Using a scientific calculator to assist in calculations).

Example 1: Solve the equation x 3+

Solution: a=1, b=, c=, d=.

a=0；b=0。

a=b=0, apply Shengjin's formula to solve, and obtain:

x⑴=x⑵=x⑶=－。

Example 2: Solve the equation 2x 3+11x 2+182x+255=0, solution: a=2, b=11, c=182, d=255.

a=－971；b=－2588；c=24709，δ=102667500。

>0, apply Shengjin's formula to solve.

y⑴=；y⑵=－33314. 49167。

Substituting the relevant values into Shengjin's formula, we get:

x⑴=－；x（2,3）=－2±9i。

Example 3: Solve the equation x 3+

Solution: a=1, b=, c=, d=.

a=；b=；c=，δ=0。

>0, apply Shengjin's formula to solve.

k=。Substituting the relevant values into Shengjin's formula, we get:

x⑴=－；x⑵=x⑶=－。

Example 4: Solve the equation 100x 3 420x 2+467x 105=0

Solution: a=100, b=420, c=467, d=105.

a=36300；b=－101640；c=85789，δ<0。

>0, apply Shengjin's formula to solve.

Substituting the relevant values into Shengjin's formula, we get:

x⑴=3/10；x⑵=5/2；x⑶=7/5。

The above results are correct after testing with Veda's theorem (process omitted).

Example 5, a building roof to build a water storage tank, according to the design requirements of the construction, the sum of the length, width and height of the water storage tank is, and width = height, the full water storage is, the three-dimensional diagonal is, Q: how to construct to meet the design requirements?

Solution: Set the length, width, and height of x dm, x dm, and x dm respectively, according to the meaning of the topic

x⑴+x⑵+x⑶=；

x⑴x⑵x⑶=；

x⑴^2+x⑵^2+x⑶^2=。

Solve this system of equations.

According to Veda's theorem, we get a cubic equation of one element:

x^3－a=1，b=－，c=，d=－。

a=289；b=－；c=，δ=0。

According to Shengjin's discriminant, this equation has three real roots, two of which are equal.

Apply Shengjin's formula to solve.

k=—。Substituting the relevant values into Shengjin's formula, we get:

x⑴=；x⑵=x⑶=。

After inspection, the results were correct.

Width = height, should be taken as long; Width = height = to carry out construction.

As long as you are proficient in operating the scientific calculator, you can easily use the Shengjin formula to solve the one-dimensional cubic equation of any real coefficient.

3. All unary cubic equations.

Root discriminant.

Total discriminant δ = b2 4ac.

When a=b=0, Sheng Jin's equation 1:

When δ=b2 4ac>0, Sheng Jin's equation 2:

Where, when δ=b2 4ac=0, Shengjin equation 3:

Thereinto. When δ=b2 4ac<0, Shengjin's equation 4: