Quadratic equations are second level equalities with one variable. They reflect the behavior of the parabola on the coordinate plane. Searched roots display the points at which the graph crosses the axis OX. According to the coefficients, you can previously find out certain qualities of the parabola. For example, if the value of the number in front of x 2 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
The school teaches several types of quadratic equations. Depending on this, there are distinguished methods of their solutions. Among the special types, quadratic equations with a parameter can be distinguished. This type contains several variables:
ah 2 + 12x-3 = 0
The following variation can be called an equation in which a variable is represented not by a single number, but by an integer expression:
21 (x + 13) 2 -17 (x + 13) -12 = 0
It is worth considering that this is all a general form of quadratic equations. It happens that they are presented in a format in which they must first be put in order, factorized or simplified.
4 (x + 26) 2 - (- 43x + 27) (7-x) = 4x
Decision principle
Quadratic equations are solved in the following way:
- If necessary, the range of acceptable values is found.
- The equation is given in the appropriate form.
- The discriminant is found by the corresponding formula: D = b 2 -4ac.
- In accordance with 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, the roots of the equation are found.
- Then (depending on the task) build a graph or find the value at a certain point.
Quadratic equations:
Vieta's theorem and other tricks
Each student wants to show off in the classroom with his knowledge, ingenuity and skills. 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 of x 1 and x 2 is equal to s. Such equations are called reduced.
x 2 -20x + 91 = 0,
x 1 * x 2 = 91 and x 1 + x 2 = 20, => x 1 = 13 and x 2 = 7
Another way to pleasantly simplify the mathematical work is to use the properties of the parameters. So, if the sum of all parameters is 0, then we get that x 1 = 1 and x 2 = s / a.
17x 2 -7x-10 = 0
17-7-10 = 0, therefore, root 1: x 1 = 1, and root 2 : x 2 = -10 / 12
If the sum of the coefficients a and c is equal to b, then x 1 = -1 and, accordingly, x 2 = -c / a
25x 2 + 49x + 24 = 0
25 + 24 = 49, therefore, x 1 = -1 and x 2 = -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 carried out in the mind, without wasting precious minutes of control or verification work on multiplying in a column or using a calculator.
Quadratic equations serve as a link between numbers and the coordinate plane. To quickly and easily build a parabola of the corresponding function, it is necessary to find a vertical line perpendicular to the x axis after finding its vertex. After that, each obtained point can be mirrored relative to a given line, which is called the axis of symmetry.