High School Mathematics Multivariate Multiple Equation Solution

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Unary quadratic equations.

You can use the direct opening method, the formula method, the matching method, and the factorization method.

Special equations are only applicable after the direct opening method and factorization method, and the matching method and formula method are suitable for all one-dimensional quadratic equations).

The multivariate quadratic equation only needs to add the idea of elimination on the basis of the one-element quadratic equation, and the specific elimination method can adopt the substitution elimination method and the addition and subtraction elimination method.

Unary cubic equations.

It can be solved by substituting the Cardano formula.

The multivariate cubic equation only needs to add the idea of elimination on the basis of the one-element cubic equation, and the specific elimination method can adopt the substitution elimination method and the addition and subtraction elimination method.

Unary quadratic equations.

It can be solved using the Ferrari solution, or it can be solved using the permutation group solution, and the specific solution of the permutation group solution method is as follows:

The multivariate quadratic equation only needs to add the idea of elimination on the basis of the unary quadratic equation, and the specific elimination method can adopt the substitution elimination method and the addition and subtraction elimination method.

For the solution of a univariate nth order equation, the numerical solution can usually be solved by iterative method by:

1. To determine the initial value of x, this equation can be taken as x0=

2. Determine the iterative formula of x, i.e.

x(k+1)=3377/175000((1+x(k))^60-1)/(1+x(k))^60

3. Then the family infiltrates 74 iterations of calculation, and we can get x=calculation error <1e-8).

1. The formula for finding the root of a cubic equation is called "Cardano's formula". The general form of a unary cubic equation is x3 + sx2 + tx + u = 0.

2. If you make an abscissa translation y=x+s 3, then the quadratic term of the equation can be eliminated. So just consider a cubic equation of the form x3=px+q.

3. Example: Suppose the solution x of the equation can be written as x=a-b, where a and b are the pending parameters.

Substituting the equation: a3-3a2b+3ab2-b3=p(a-b)+q

The whole is obtained: a3-b3=(a-b)(p+3ab)+q;From the theory of quadratic equations, it can be seen that a and b must be appropriately selected, such that at the same time as x=a-b, 3ab+p=0. In this way, the above equation becomes a3-b3=q, multiplied by 27a3 on both sides, and 27a6-27a3b3=27qa3.

From p=-3ab, we can see that 27a6+p=27qa3 is a quadratic equation about a3, so a can be solved.

The solution of the unary higher-order equation is as follows:

1) Cubes x=a (1 3) + b (1 3) at the same time.

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.

4) x 3 3(ab) (1 3) x (a+b) 0, compared with the unary cubic equation and the special type x 3+px+q=0.

5）－3(ab）^（1/3）＝p,－(a+b)=q。

6）a+b＝－q，ab＝-（p/3）^3。

7) This is in fact a formulation of the root finding of a cubic equation into a formula for finding the root of a quadratic equation, because a and b can be regarded as the two roots of a quadratic equation, and (6) is the Vedic theorem of the form ay 2+by+c=0.

8）y1＋y2＝－（b/a）,y1*y2=c/a。

9) Compare (6) and (8) to make a y1, b y2, q b a, -(p 3) 3 c a.

1. Estimation method: the introductory method when you have just learned to solve equations. The solution of the equation is directly estimated and then verified by substitute for the original equation.

2. Apply the properties of the equation to solve the equation.

3. Merge similar terms: deform the equation into a monomial.

4. Move terms: Move the terms with unknown numbers to the left and the constant terms to the right.

For example: 3+x=18

Solution: x=18-3

x=155, remove brackets: Use the deparentheses rule to remove the brackets in the equation.

4x+2（79-x）=192

Solution: 4x+158-2x=192

4x-2x+158=192

2x+158=192

2x=192-158

x=176, formula method: there are some equations, the general form of the solution has been studied, and it has become a fixed formula, which can be directly used by the formula. Solvable multivariate higher-order equations generally have formulas to follow.

7. Function image method: The solution of the equation is used to solve the geometric meaning of the intersection of two or more related function images.

Extended Materials. Basis for solving the equation.

1. Shift term change: move some terms in the equation from one side of the equation to the other side with the previous symbols, and add, subtract, subtract, multiply and divide, divide by multiplication;

2. The basic properties of the equation.

Property 1: The same number or the same algebraic formula is added (or subtracted) to both sides of the equation at the same time, and the result is still the equation. It is expressed in letters as: if a=b, c is a number or an algebraic formula.

1)a+c=b+c

2)a-c=b-c

Property 2: If both sides of the equation are multiplied or divided by the same non-0 number, the result will still be the equation.

It is expressed in letters as: if a=b, c is a number or an algebraic formula (not 0). Then:

a c = b c or a c = b c

Property 3: If a=b, then b=a (symmetry of the equation).

Property 4: If a = b, b = c then a = c (transitivity of the equation).

The fast solution of the high school unary cubic equation is as follows:

There is no fast solution to the unary cubic equation, and there is the famous Caldan formula for solving the unary cubic equation with the root number, but the solution using the Caldan formula is more complicated and lacks intuitiveness. Fan Shengjin deduced a set of new root-finding formulas for the general form of unary cubic equations expressed directly in a, b, c, and d: Shengjin's formula.

Shengjin's theorem: when b=0, c=0, Shengjin's formula 1 is meaningless; When a=0, Shengjin's equation 3 is meaningless; When a 0, Shengjin's equation 4 is meaningless; When t<1 or t>1, Shengjin's equation 4 is meaningless.

When b=0, c=0, does Shengjin's equation 1 hold? Is there a value of a 0 in Shengjin Formula 3 and Shengjin Formula 4? Does Shengjin's formula 4 have a value of t< 1 or t>1? Shengjin's theorem gives the following:

Shengjin's theorem 1: When a=b=0, if b=0, then there must be c=d=0 (in this case, the equation has a triple real root 0, and Shengjin's equation 1 still holds).

Shengjin's theorem 2: When a=b=0, if b≠0, then there must be c≠0 (in this case, Shengjin's formula 1 is applied to solve the problem).

Shengjin's theorem 3: When a=b=0, then there must be c=0 (in this case, Shengjin's formula 1 is applied to solve the problem).

Shengjin's theorem 4: When a=0, if b≠0, then there must be δ>0 (in this case, Shengjin's formula 2 is applied to solve the problem).

Shengjin's theorem 5: When a<0, there must be δ>0 (in this case, Shengjin's formula 2 is applied to solve the problem).

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In middle school (including high school), the solution of unary cubic equations is not taught. However, he can use the knowledge of factorization to solve some special one-dimensional cubic equations.

The standard form of a cubic equation (i.e., the form in which all cubic equations can be collated): ax3+bx2+cx+d=0 (a, b, c, d are constants, x is unknown, and a≠0). The formula solutions of the unary cubic equation include the Caldan Gong imitation method and the Shengjin formula method.

Once, there is no need to talk about quadratic equations.

Cubic equations have a formula for finding the root (Cardin's formula for empty height).

Quadratic equations have a formula for finding the root (Ferrari's formula).

Special equations of five or more degrees, such as the binomial equation (x n=a), have a formula for finding the roots, and the root is connected to the failure, and all the roots are obtained.

There is no formula for finding the root of a general equation of five or more, but an equation of real coefficients must be decomposed into the product of the first factor of the real coefficient and the quadratic factor of the real coefficient. Numerical solutions are usually used. For odd-order equations, because they have at least one real root, this real root can be obtained by methods such as the dichotomy method, and the equation can be reduced.

For even-numbered equations, there may not be real roots, and the Linsberger-Zhao method (hidden tremor splitting factor method) is commonly used to iteratively find a real quadratic factor of the equation, so that the equation can also be reduced (of course, this method is also applicable to odd-order equations).From this, all the roots of the equation can be found.