Quadric equations are second level equalities with one variable. They reflect the behavior of the parabola on the coordinate plane. The desired roots display the points at which the graph intersects the OX axis. By the coefficients, you can first find out certain qualities of the parabola. For example, if the value of the number before x2 is negative, then the branches of the parabola will look up. In addition, there are several tricks with which you can significantly simplify the solution of a given equation.
Types of quadratic equations
Several types of quadratic equations are taught at school. Depending on this, there are also ways to solve them. Among the special types, quadratic equations with a parameter can be distinguished. This type contains several variables:
ah2+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 thiseverything is a general form of quadratic equations. Sometimes they are presented in a format in which they must first be put in order, factored or simplified.
4(x+26)2-(-43x+27)(7-x)=4x
Decision principle
Quadric equations are solved in the following way:
- If necessary, find the range of acceptable values.
- The equation is given in the appropriate form.
- The discriminant is found according to the corresponding formula: D=b2-4ac.
- According to the value of the discriminant, conclusions are drawn regarding the function. If D>0, then they say that the equation has two different roots (for D).
- After that, find the roots of the equation.
- Next (depending on the task) build a graph or find the value at a certain point.
Quadric equations: Vieta's theorem and other tricks
Each student wants to show off his knowledge, ingenuity and skills in the classroom. While studying quadratic equations, this can be done in several ways.
In the case when the coefficient a=1, we can talk about the application of the Vieta theorem, according to which the sum of the roots is equal to the value of the number b in front of x (with a sign opposite to the existing one), and the product x 1 and x2 is equal to c. Such equations are called reduced.
x2-20x+91=0,
x1x2=91 and x1+x 2=20,=> x1=13 and x2=7
MoreOne way to nicely simplify the math work is to use the properties of the parameters. So, if the sum of all parameters is 0, then we get that x1=1 and x2=c/a.
17x2-7x-10=0
17-7-10=0, therefore root 1: x1=1, and root 2: x2=- 10/12
If the sum of the coefficients a and c is equal to 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 equations greatly simplifies the calculation process, and also saves a huge amount of time. All actions can be performed in the mind, without spending precious minutes of control or verification work on multiplication in a column or using a calculator.
Quadric equations serve as a link between the numbers and the coordinate plane. To quickly and easily construct a parabola of the corresponding function, it is necessary, after finding its vertex, to draw a vertical line perpendicular to the x-axis. After that, each obtained point can be mirrored relative to a given line, which is called the axis of symmetry.
