Equations in mathematics are just as important as verbs in Russian. Without the ability to find the root of the equation, it is difficult to assert that the student has learned the course of algebra. In addition, for each of their species, there are special ways to solve.
What it is?
An equation is two arbitrary expressions containing variable quantities, between which an equal sign is placed. And the number of unknown quantities can be arbitrary. The minimum number is one.
To solve it is to know if there is a root of the equation. That is, a number that turns it into a true equality. If it is not, then the answer is the statement that "there are no roots". But there can be the opposite, when the answer is a set of numbers.
What types of equations are there?
Linear. It contains a variable whose degree is one.
- Square. Is variable with degree 2, or convert the lead to an extent.
- The 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 the linear equation solved?
It is basic. To this kind all others want to lead. Since it is quite easy to find the root of the equation.
- First, you need to perform possible transformations, that is, to open parentheses and to bring similar terms.
- Transfer all monomials with variable value to the left side of the equation, leaving the free terms in the right side.
- Give similar terms in each part of the solved equation.
- 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 must be brought to the standard form, that is, to reveal all the parentheses, to bring up such terms and to transfer all the monomials to the left. On the right side of the equation, only zero must 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. He is negative - the decision is over, since he has no roots. Is equal to zero - the answer is one number. Positive - two values for 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 numbers that are divisors of the free term. It is convenient to consider this method on a concrete example. Let the equation have the form: x 3 - 3x 2 - 4x + 12 = 0.
Its free term is 12. Then the dividers to be checked are positive and negative numbers: 1, 2, 3, 4, 6 and 12. The search can be completed already on the number 2. It gives the right equality in the equation. That is, its left side is equal to 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 numbers: 3 and -2.
In summary, the concrete cubic equation has three roots: 2, -2, and 3.
How are systems of linear equations solved?
Here we propose a method for eliminating unknowns. It consists in expressing one unknown through another in one equation and substituting this expression for another. And the solution of a system of two equations with two unknowns is always a pair of variables.
If in them the variables are denoted by the letters x1 their2. then we can derive from the first equality, for example, x2. Then it is substituted in the second. The necessary transformation is carried out: the disclosure of parentheses and the reduction of such 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 response received, it is recommended to always check. It does not need to be recorded.
If one equation is solved, then each of its roots must be substituted into the original equation and get the same numbers in both its parts. All came together - the solution is right.
When working with the system, the roots need to be substituted into each solution and perform all possible actions. Is the right equality? So the solution is correct.