Instruction

1

Make a note of the original cubic equation in the form: x3+a*x2+b*x+c=0. To do this, all the coefficients

**of the equation**divide by the first coefficient with a multiplier of x3 so that it became equal to one.2

Based on the algorithm of the method of vieta, cardan, calculate the values of R and Q according to the respective formulae: Q =(a2-3b)/9, R=(2a3-9ab+27c)/54. Moreover, the coefficients a, b and C are the coefficients of the given

**equation**.3

Compare the obtained values of R and Q. If it is the expression Q3 >R2 , therefore, in the initial equation there are 3 real roots. Calculate them by the formulas of vieta.

4

When Q3 values <= R2 , the solution is one real root x1 and two complex conjugate roots. To define them we need to find intermediate values A and B. Calculate their formulas Cardano.

5

Find the first valid root by the formula x1=(B + A) a/3. For different values of A and b define the complex conjugate roots of the cubic

**equation**by appropriate formulas.6

If the values of A and b was equal, conjugate roots, degenerate in the second real root of the original

**equation**. This is the case when a valid root turns two. Calculate second real root by the formula x2=-A-A/3.# Advice 2: How to solve equations

The solution of the equations , without which it is impossible to do in physics, mathematics, chemistry. At the very least. Learn the basics of their decision.

Instruction

1

In the most General and simple classification of the equation can be divided by the number of variables they contain and the degree in which these variables are.

To solve an equation means to find all its roots or to prove that they are not.

Any of the equations has at most P roots, where P is the maximum degree of a given equation.

But some of these roots may coincide. For example, the equation x^2+2*x+1=0, where ^ is the icon exponentiation, folded in a square of expression (x+1), that is the product of two identical brackets, each of which gives x=-1 as solutions.

To solve an equation means to find all its roots or to prove that they are not.

Any of the equations has at most P roots, where P is the maximum degree of a given equation.

But some of these roots may coincide. For example, the equation x^2+2*x+1=0, where ^ is the icon exponentiation, folded in a square of expression (x+1), that is the product of two identical brackets, each of which gives x=-1 as solutions.

2

If the equation has only one unknown, it means that you will be able to explicitly find the roots (real or complex).

To do this, most likely need different transformations: formulas of reduced multiplication to the calculation formula of the discriminant and roots of a quadratic equation, the transfer of terms from one part to another, the bringing to a common denominator, multiply both sides by the same expression, squaring, etc.

Conversion, not affecting the roots of the equation are called identical. They are used to simplify the solution of the equation.

You can also instead of the traditional use of analytical graphical method and write the given equation as a function, then conducting her research.

To do this, most likely need different transformations: formulas of reduced multiplication to the calculation formula of the discriminant and roots of a quadratic equation, the transfer of terms from one part to another, the bringing to a common denominator, multiply both sides by the same expression, squaring, etc.

Conversion, not affecting the roots of the equation are called identical. They are used to simplify the solution of the equation.

You can also instead of the traditional use of analytical graphical method and write the given equation as a function, then conducting her research.

3

If the equation more than one unknown, you can only Express one of them through the other, thereby showing the set of solutions. Such, for example, equations with parameters, which contain the unknown x and the parameter a. To solve the parametric equation means for all and to Express x in a, that is, to consider all possible cases.

If in the equation are derivatives or differentials of an unknown (see picture), congratulations, it's a differential equation, and then you can not do without mathematics).

If in the equation are derivatives or differentials of an unknown (see picture), congratulations, it's a differential equation, and then you can not do without mathematics).

# Advice 3: How to solve cubic equation

Today, the world knows several ways to solve a cubic equation. The most popular are the formula of Cardan and trigonometric formula of vieta. However, these methods are rather complicated and in practice almost never used. The following is the easiest way to solve the cubic equation.

Instruction

1

So, in order to solve the cubic equation of the form Ах3+Вх2+CX+D=0, you need brute force to find one of the roots of the equation. The root of the cubic equation is always one of the divisors of the free term of the equation. Thus, in the first stage the solution of the equation, you need to find all integers a for which the constant D is divisible.

2

Received integers are alternately substituted into the cubic equation instead of the unknown variable x. The number that draws the equality of the faithful, is the root of the equation.

3

One of the roots of the equation found. For further solutions to apply the method of dividing a polynomial of the binomials. Polynomial Ах3+Вх2+CX+D is divisible, and the binomials x-h where h - the first root of the equation XX). The result of the division will be a square polynomial of the form ах2+bx+C.

4

Equating the obtained polynomial to zero ах2+bx+C =0, you get a quadratic equation, the roots of which and will be the solution to the original cubic equation, i.e. x₂' ₃ ơ=(-b±√(b^2-4ac))/2a

Note

In the first stage equation, namely, finding the root of equation by the method of selection, we should not forget about the whole negative numbers, which can also be a solution of the equation.