Solving quadratic equations and constructing graphs

The quadratic equations are equalities of the secondlevel with one variable. They reflect the behavior of the parabola on the coordinate plane. The required roots represent the points at which the graph intersects the OX axis. By coefficients one can first know certain qualities of a parabola. For example, if the value of the number before x², the negative, then the branches of the parabola will look up. In addition, there are several tricks with which you can greatly simplify the solution of a given equation.

Types of quadratic equations

At school are taught several kinds of squareequations. Depending on this, the methods of their solutions are also differentiated. Among the special types, one can single out quadratic equations with a parameter. This type contains several variables:

Oh²+ 12x-3 = 0

The next variation is an equation in which the variable is represented not by a single number but by an entire expression:

21 (x + 13)²-17 (x + 13) -12 = 0

It is worth considering that this is all a general viewsquare equations. Sometimes they are presented in a format in which they must first be put in order, multiplied or simplified.

4 (x + 26)²- (- 43x + 27) (7-x) = 4x

Principle of solution

Quadratic equations are solved in the following way:

1. If necessary, there is an area of ​​acceptable values.
2. The equation is reduced to the corresponding form.
3. There is a discriminant according to the corresponding formula: A = b²-4ас.
4. In accordance with the value of the discriminant, conclusions are made about the function. If A> 0, then we say that the equation has two distinct roots (for A).
5. After this, the roots of the equation are found.
6. Further (depending on the task), a graph is plotted or a value is found at a certain point.

Square equations: Vieta's theorem and other tweaks

Each schoolchild wants to shine in the classroom with his knowledge, ingenuity and skills. During the study of quadratic equations, this can be done in several ways.

In the case when the coefficient a = 1, we canto talk about the application of Viet's theorem, according to which the sum of the roots is equal to the value of the number b, which is in front of x (with a sign opposite to the existing one), and the product x₁ their₂ is equated with. Such equations are called reduced.

x²-20x + 91 = 0,

x_1*x₂= 91 and x₁+ x₂= 20, => x₁= 13 and x₂= 7

Another way to pleasantly simplify mathematical work is to use the properties of parameters. So, if the sum of all the parameters is 0, then we get that x₁= 1 and x₂= c / a.

17x²-7x-10 = 0

17-7-10 = 0, therefore, the root 1: x₁= 1, and the root z: x₂= -10 / 12

If the sum of the coefficients a and c is b, then x₁= -1 and, respectively, x₂= -c / a

25x²+ 49x + 24 = 0

25 + 24 = 49, therefore, x₁= -1 and x₂= -24 / 25

This approach to solving quadratic equationsgreatly simplifies the calculation process, and also saves a huge amount of time. All actions can be carried out in the mind without spending precious minutes of control or verification work on multiplication in a column or using a calculator.

Square equations serve as a linkbetween the digits and the coordinate plane. To quickly and easily construct a parabola of the corresponding function, after finding its vertex, it is necessary to draw a vertical line perpendicular to the x axis. After this, each obtained point can be mirrored with respect to a given line, which is called the axis of symmetry.

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