Quadratic equations are the equations of the second level with a single variable.They reflect the behavior of a parabola on the coordinate plane.The unknown roots represent points where the graph crosses the x-axis.By factors can be found pre-defined quality of the parabola.For example, if the number of standing in front of x2 is negative, the branches of the parabola will look up.In addition, there are a few tricks that you can use to simplify the solution of the given equation.
Types quadratic equations
The school taught several types of quadratic equations.Depending on this distinction and solutions.Among the special types can be distinguished quadratic equations with a parameter.This type contains a number of variables:
ax2 + 12X-3 = 0
Another variation can be called an equation in which the variable is not represented by a single number, and the whole expression:
21 (x + 13) 2-17 (x13) -12 = 0
It's worth noting that this is the common view of all quadratic equations.Often, they are presented in a format in which they must first be put in order, to factor and simplify.
4 (x + 26) 2 - (- 43h + 27) (7 x) = 4
principle solutions
Quadratic equations are solved in the following manner:
- If necessary, is the range of permissible values.
- equation leads to the appropriate type.
- Located on the discriminant of the formula: A = b2-4as.
- In accordance with the value of the discriminant conclusions about the function.If L & gt; 0, then we say that the equation has two different roots (at D).
- Then find the roots of the equation.Further
- (depending on the assignment) are plotted or value at a certain point.
Quadratic equations: Vieta theorem and other tricks
each student wants to shine at the lessons of their knowledge, skills and acumen.During the study of quadratic equations it can be done in several ways.
In the case where the coefficient a = 1, we can talk about the use of Wyeth's theorem, according to which the sum of the roots is equal to the value of b, standing in front of x (with a sign opposite is available), and the product of x1 and x2 is equal to.Such equations are called forth.
h2-20h + 91 = 0,
x1 * x2 = 91 and x1 + x2 = 20, = & gt;x1 = 13 and x2 = 7
Another pleasant way to simplify the mathematical work is to use the properties settings.So, if the sum of all parameters is 0, it follows that x1 = 1 and x2 = c / a.
17h2-7h-10 = 0
17-7-10 = 0, therefore, the root of 1: x1 = 1 and koren2 x2 = -10/12
If the sum of the coefficients a and c is equal to b, thenx1 = -1 and accordingly, x2 = c / a
25h2 + 49h + 24 = 0
25 + 24 = 49, therefore, x1 = -1 and x2 = -24 / 25
This approach to the solution ofquadratic equations significantly simplifies the calculation process, and saves huge amounts of time.All actions can be done in the mind, without spending precious moments of control or verification work on multiplication in the column or use a calculator.
Quadratic equations serve as a link between numbers and the coordinate plane.To quickly and easily build a parabola corresponding function, it is necessary after finding its top draw a vertical line perpendicular to the x-axis.Thereafter, each point can be obtained with respect to mirror a given line, which is called the axis of symmetry.
