I have a formula for finding the root of a cubic third and fourth order equation

The multivariate formula has changed...

1) The content of the Vedic theorem.

The unary quadratic equation ax 2+bx+c=0 (a≠0 and =b 2-4ac 0).

Let the two roots be x1 and x2

Then x1+x2= -b a

x1*x2=c/a

2) Generalization of the Vedic theorem.

The Vedic theorem can also be used in higher order equations. In general, for a unary n-order equation aix i=0

Its roots are denoted as x1, x2....,xn

We have. xi=(-1)^1*a(n-1)/a(n)xixj=(-1)^2*a(n-2)/a(n)xi=(-1)^n*a(0)/a(n)

where is the sum and is the product.

That's right, it's just that after 3 times, there is no such relationship, and even after 5 times, there is no corresponding root finding formula. Thank you...

The formula for finding the root of a quadratic equation is as follows:

ax4+bx3+cx2+dx+e=0 (a,b,c,d,e r, and a is the superscript number).

The four roots of the property equation are:

x1=(-b+a+b+k)/(4a)

x2=(-b-a+b-k)/(4a)

x3=(-b+a-b-k)/(4a)

x4=(-b-a-b+k)/(4a)

The letters a, b, and k are sufficient to represent any three complex numbers, according to Veda's theorem: the sum of the four roots of the equation is -b a, so when the algebraic formula of x1, x2, and x3 is the three roots of the original equation, then the algebraic formula of x4 must be the fourth root of the equation. )

Substituting these four algebraic formulas into Veda's theorem yields the following: x1+ x2+ x3+ x4= -b ax1x2 +x1x3+ x1x4+ x 2 x3 + x2x4+ x3 x4=(1 8a2)(3b2-a2-b2-k2)=c ax1x2x3 +x1x2x4+ x1 x3 x4+ x2 x3 x4= (1 16a3)(-b3+ba2+bb2+bk2+2abk)= d a

x1x2 x3 x4=(1/256a4)(b4+ a4+b4+k4-2b2a2-2b2b2-2b2k2-2a2b2-2a2k2-2b2k2-8babk)=e/a

<> formula for finding the root of a unary cubic equation is ax3+bx2+cx+d=0, i.e., ax 3+bx 2+cx+d=0 (a, b, c, d belong to r, x is unknown, and a is not equal to 0) The equation refers to an equation containing unknowns. It is an equation that expresses the equality relationship between two mathematical formulas (such as two numbers, functions, quantities, operations), and the value of the unknown number that makes the equation hold is called the solution or root. The process of finding the solution of an equation is called solving an equation.

By solving the equation, you can avoid the difficulty of reverse thinking, and directly list the equation containing the quantity you want to solve. Equations have a variety of forms, such as unary linear equations, binary linear equations, unary quadratic equations, etc., and can also form a system of equations to solve multiple unknowns.

The universal simplification formula of a cubic equation is ax3+bx2+cx+d=0, and the cubic equation contains only one unknown (i.e., "yuan"), and the highest number of unknowns is an integer equation of the 3rd degree.

Ordinary cubic equations cannot be solved by the fitting method, but quadratic equations can. The standard solution of quadratic equations is to square both sides of the equation after introducing the parameters, and then solve them by opening the square on both sides, and the parameters are obtained by solving a cubic equation. There are only square roots and cubic roots in the root finding formula of the obtained quadratic equation, and there is no quadratic root, so the solution of the quadratic equation can also be directly calculated by calculating the square and the square of the quadratic equation.

Equation solution: 1. Caldan formula published by the Italian scholar Caldan in 1545;

2. The Shengjin formula published by Chinese scholar Fan Shengjin in 1989.

Both formula methods can solve standard one-dimensional cubic equations. It is convenient to use the Caldan formula to solve the problem, in contrast, although the Shengjin formula is simple in form, it is more lengthy and inconvenient to memorize, but the actual solution is more intuitive.

When δ = (q 2)2

p 3) At 30, the excavation equation has a solid root and a pair of conjugate virtual roots.

When δ = (q 2)2

p 3) At 30, the equation has three real roots, one of which has a double root;

When δ = (q 2)2

p 3) At 30, the equation has three unequal real roots.

When δ=b24ac 0, x=[-b (b24ac)( 2a when δ=b2

At 4ac 0, x = 2a

You want a one-dimensional formula for finding roots, right? The solution method of the root finding formula of the unary cubic equation The root finding formula of the unary cubic equation cannot be made by ordinary deductive thinking, and the matching method similar to the root finding formula of the unary quadratic equation can only formalize the standard one-dimensional cubic equation of type ax 3+bx 2+cx+d+0 into the special type of the destruction of the fiber with the roll x 3+px+q=0. The solution of the formula for solving a cubic equation can only be obtained by inductive thinking, that is, the form of the formula for finding the root of a cubic equation is summarized according to the form of the root finding formula of a one-dimensional cubic equation, a one-dimensional quadratic equation and a special higher-order equation.

The root finding formula of a cubic equation of the form x 3+px+q=0 should be of the form x=a (1 3)+b (1 3), that is, the sum of the two open squares. After summarizing the form of the formula for finding the root of a cubic equation, the next step is to find the contents of the cube, that is, to use p and q to represent a and b. Here's how:

1) Cubes x=a (1 3)+b (1 3) at the same time to get (2) x 3=(a+b)+3(ab) (1 3)(a (1 3)+b (1 3)) 3) Since x=a (1 3)+b (1 3), (2) can be reduced to x 3=(a+b)+3(ab) (1 3)x, and the shift term can obtain (4) x 3 3(ab) (1 3)x (a+b) 0, and the unary cubic equation and the special type x 3+px+q=0 are compared with the sale, It can be seen that (5) 3(ab) (1 3) p, (a+b)=q, simplified to (6)a+b q,ab -(p 3) 3 (7) In this way, the root finding formula of a cubic equation is actually reduced to the root finding formula of a quadratic equation, because a and b can be regarded as a quadratic equation.