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 2: How to solve equation third degree

Equations of the third degree is also called cubic equations. This equation, in which the highest degree when variable x is a cube (3).

Instruction

1

Cubic equation in General form looks like: ax3 + bx2 + cx + d = 0, a does not equal 0; a, b, c, d are real numbers. A universal method of solving equations of the third degree method is Cardano.

2

For starters, here is the equation to the form y3 + py + q = 0. To do this, replace the variable x by y - b/3a. Substitution replacement, see in the picture. To disclose the brackets used two formulas of reduced multiplication: (a-b)3 = a3 - 3a2b + 3ab2 - b3 and (a-b)2 = a2 - 2ab + b2. Then, given similar terms and grouped by powers of the variable y.

3

Now, to get under y3 unit ratio, divide all equation on a. We obtain the following formulas for the coefficients p and q in the equation y3 + py + q = 0.

4

Then computed the special values: Q, α, β, which will allow to calculate the roots of the equation with y.

5

Then the three roots of the equation y3 + py + q = 0 are calculated by the formulas in the figure.

6

If Q > 0, the equation y3 + py + q = 0 has only one real root y1 = α + β (and two complex, evaluate them according to prescribed formulas, if necessary).

If Q = 0, all roots real and at least two of them coincide, with α = β and equal roots: y1 = 2α, y2 = y3 = -α.

If Q < 0 then the roots are real, but you need the ability to extract the root of a negative number.

After finding y1, y2 and y3, substitute them into the substitution x = y - b/3a and find the roots of the original equation.

If Q = 0, all roots real and at least two of them coincide, with α = β and equal roots: y1 = 2α, y2 = y3 = -α.

If Q < 0 then the roots are real, but you need the ability to extract the root of a negative number.

After finding y1, y2 and y3, substitute them into the substitution x = y - b/3a and find the roots of the original equation.

Is the advice useful?

If you can find one of the roots of the cubic equation x1, it can be a cubic polynomial divided by (x - x1) and solve the resulting quadratic equation.