How to find the double root of y 3 1 0? 20

The New4ton method of solving the nonlinear square 3 way f(x)=0 is a y approximate square 6 method that linearizes the nonlinear square 7 to 7. O f(x) near x0 to a Taylor series f(x) = f(x0)+(x x0)f'(x0)+(x－x0)^2*f''(x0)。2!

Take its linear part 5 as the approximate square 1 of 1 nonlinear square 6 degrees f(x) = 0, that is, the first two terms of Taylor y, then there is f(x0)+f'(x0)(x x0)=f(x)=0 Let f'(x0)≠0 then its solution is 7x4=x0 f(x0). f'(x0) In this way, we get an e m iteration sequence of New3ton's method: x(n+8)=x(n) f(x(n)).

f'(x(n))。For the problem of uf(x)=0 with multiple roots, the following m formula x(n+4)=x(n) f(x(n))*f should be used'(x(n))。f'(x(n))）2-f(x(n))*f''(x(n))], while finding the complex root is +i after the initial value

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1,1 2+2 points root number 3i,1 2-2 points root number 3i; Three in total.

You can simply assume that the root is a+bi and solve the system of real equations.

That is, a3-3ab2=-1, 3a2b-b3=0

Solution constructor f(x)=x 3-x+

then f'(x)=3x^2-1

Order f'(x)=0

Solution x= root 3 3

Then when x=-root3 3, the function has a maximum value f(-root3 3) 0, so the image of f(x) has only one intersection point, then the equation.

x 3-x+ has only one real root, and the modern algebra principle knows that the equation x 3-x+ has two complex roots.

The x-x+ solution is one solid root and two complex roots. x=

x^3+1

x+1)(x^2-x+1)

The two complex roots are two roots of x 2-x+1=0, and x=(1 5i) 2.

Answer: Complex numbers where imaginary parts are opposites of each other are conjugate complex numbers.

A solution to a quadratic equation with an imaginary solution.

It is a pair of conjugated reroots.

The figure is not clear, but the conjugate double root is probably i*i = -1; See.

According to the formula for finding the root of a quadratic equation, Vedic theorem:

Finding the conjugate complex root is usually met with a discriminant formula less than 0In the range of real numbers there is no solution, while in the range of complex numbers because the square of i = -1So, just do this by doing this with the number that is originally less than in the root number.

For example, if the root number is -1, then it is +i and -i.

The general form of a quadratic equation is as follows:

Determine the discriminant formula and calculate the δ (

Greek letters, transliterated as Delta).

If 0 is δ>, the equation has two unequal real roots in the field of real numbers: ;

If δ=0, the equation has two equal real roots in the real number field:

If δ< 0, the equation has no solution in the real number domain, but there are two conjugate complex roots in the imaginary number field, which isThe concept of imaginary numbersIn mathematics, an imaginary number is a number of the form a+b*i, where a, b are real numbers, and b≠0, i = - 1. The term imaginary number was coined by the famous mathematician Descartes in the 17th century, because the concept at the time was that it was a number that did not really exist. Later, it was found that the real part a of the imaginary number a+b*i corresponds to the horizontal axis on the plane, and the imaginary part b corresponds to the vertical axis on the corresponding plane, so that the imaginary number a+b*i can correspond to the points (a, b) in the plane.

The imaginary number bi can be added to the real number a to form a complex number of the form a + bi, where the real numbers a and b are called the real and imaginary parts of the complex number, respectively. Some authors use the term pure imaginary number to denote the so-called imaginary number, which denotes any complex number with a non-zero imaginary part.

Conjugate plural conceptConjugate complex numbers, where two real parts are equal and imaginary parts are opposite to each other, are conjugate complex numbers. When the imaginary part is not zero, the conjugate complex is equal to the real part, and the imaginary part is the opposite, if the imaginary part is zero, its conjugate complex is itself. (When the imaginary part is not equal to 0, it is also called a conjugate imaginary number) The conjugate complex number of the complex number z is denoted as z.

At the same time, the complex number z is called the complex conjugate of the complex number z

Conjugated compound roots are a pair of special roots. Refers to a polynomial or algeon.

A class of roots that appear in pairs of number equations. If the non-real complex number is the root of the real coefficient n degree of the equation f(x)=0, then its conjugate complex * is also the root of the equation f(x)=0 and is the same as the multiple number of *, then and * are said to be a pair of conjugate complex (imaginary) roots of the equation.

The conjugate complex root often appears in a quadratic equation, and if the discriminant formula of the root is less than zero by the formula method, the root of the equation is a pair of conjugate complex roots.

Use the matching method. Conjugate the root of the law. First method:

With method b 2-4ac=-36, right? -36=(6i) 2, right? So let's substitute the root-finding formula:

2 A parts of the negative B plus or minus the root number B square minus 4 ACThe second way: let r=a+bi, and calculate it by substitution.

If it follows the complex root and b equals 0, then x1 becomes. The last else should be changed to :

x1 = - b / (2 * a);

x2 = sqrt(-delt) / (2 * a);

x2 = (x2 > 0) ?x2 : x2;

if(b!=0)

else

Use the "Solver Solution" function in Excel.

See for details.

I know that ABC is a decimal.

If you don't know ABC, it's a mathematical formula, and a cubic equation doesn't know what you want to do.

Then you can first identify Delta, and then make a formula of x1 x2 x3 depending on the situation.

Excel 2007 can be done like this:

Enter 1 in A1 and the formula in B1

a*a1^3+b*a1+c

After confirmation, select b1, Data Data Tools tab Hypothetical Analysis Univariate Solving The target cell is b1 The target value is 0 The variable cell is a1 After the determination is made, a1 is the required value.

a, b, c are variables, can be generated directly in cells (e.g. a2, a3, a4), and then a, b, c in b1 can be replaced by cells, e.g. =a2*a1 3+a3*a1+a4