Equations in mathematics are as important as verbs in Russian. Without the ability to find the root of the equation, it is difficult to say that the student has mastered the course of algebra. In addition, for each of their species there are their own specific solutions.
What it is?
An equation is two arbitrary expressions containing variables with an equal sign between them. Moreover, the number of unknown quantities can be arbitrary. The minimum number is one.
To solve it is to find out if there is a root of the equation. That is a number that turns it into the correct equality. If not, then the answer is the statement that "there are no roots." But it can be the opposite when the answer is a multitude of numbers.
What types of equations are there?
Linear. It contains a variable whose degree is equal to one.
- Square. Is variable with degree 2, or convert the lead to an extent.
- Equation of the highest degree.
- Fractional rational. When the variable is in the denominator of the fraction.
- Irrational. That is, one that contains an algebraic root.
How is a linear equation solved?
It is the main one. To this view all others seek to bring. Since it is easy to find the root of the equation.
- First, you need to perform possible transformations, that is, open the brackets and give similar terms.
- Move all monomials with variable to the left side of the equation, leaving the free terms on the right.
- Bring similar terms in each part of the equation to be solved.
- In the resulting equality in the left half will be the product of the coefficient and variable, and the right number.
- It remains to find the root of the equation by dividing the number on the right by the coefficient before the unknown.
How to find the roots of a quadratic equation?
First, it needs to be reduced to a standard form, that is, to open all the brackets, bring such terms and move all monomials to the left side. On the right side of the equality, only zero should remain.
- Use the formula for the discriminant. Erect in the square of the coefficient before the unknown with a degree "1". Free multiply a single term and the number before the variable in the square with the number 4. From the resulting square subtract work.
- Estimate the value of the discriminant. It is negative - the decision is finished, as it has no roots. Equal to zero - the answer is one number. Positive - two values of the variable.
Find the two roots of the equation according to the formula in which the square root of the discriminant is necessary to subtract or folded with a negative coefficient of the variable in the first degree. Then divided by twice the coefficient before the square is unknown. (In case of equality of the discriminant to zero to add or subtract should be zero, so the two roots coincide.)
How to solve a cubic equation?
First find the root of the equation x. It is determined by the method of selection from the numbers that are the divisors of the free term. This method is convenient to consider a specific example. Let the equation be: x 3 - 3x 2 - 4x + 12 = 0.
Its free term is 12. Then the divisors that need to be checked will be positive and negative numbers: 1, 2, 3, 4, 6 and 12. The search can be completed on the number 2. It gives the correct equality in the equation. That is, its left side turns out to be zero. So the number 2 is the first root of the cubic equation.
Now you must split the original equation into the difference variable and the first root. In the specific example is (x – 2). A simple conversion causes the numerator to this factorization: (x – 2)(x 2)(x – 3). The same factors of the numerator and denominator are reduced, and the remaining two brackets when solving a simple quadratic equation: x 2 – x – 6 = 0.
Here find the two roots of the equation according to the principle described in the previous section. They are the numbers: 3 and -2.
So, a concrete cubic equation has three roots: 2, -2 and 3.
How are linear equation systems solved?
Here a method for eliminating unknowns is proposed. It consists in expressing one unknown through the other in one equation and substituting this expression into another. Moreover, the solution of a system of two equations with two unknowns is always a pair of variables.
If in them variables are designated by letters x1 their2. then we can derive from the first equation, for example, x2. Then it is substituted into the second. The necessary transformation is carried out: disclosure of brackets and coercion of similar members. It turns out a simple linear equation, the root of which is easy to calculate.
Now go back to the first equation and find the root of the equation x2. using the resulting equality. These two numbers are the answer.
In order to be sure of the answer received, it is recommended to always make a check. It is not necessary to write.
If one equation is solved, then each of its roots must be substituted into the original equality and get the same numbers in both its parts. It all came together - the right decision.
When working with the system, it is necessary to substitute the roots in each solution and perform all possible actions. It turns out the right equality? So the decision is correct.