One of the methods used to find the results of a second degree equation and the Bhaskara's formula. The use of this formula is usually divided into two steps: the first is to find the value of the discriminating gives equation and the second in finding your results.
But what is "Discriminant"?
discriminating it is the part of the formula of Bhaskara which is under the square root.
The calculation of discriminating is done by substituting the values of the coefficients of the equation in the following formula:
Δ = b2 – 4ac
From this value, just replace it, with the coefficientsgivesequation, in the formula:
x = – b ± √Δ
2nd
The separation of this method into two steps is just didactic. THE formulainBhaskara can also be written:
x = – b ± √[b2 – 4ac]
2nd
There are other uses for the discriminating of a equationofseconddegree. Next, we'll talk about them.
Number of solutions of a quadratic equation
It may often be necessary to know if a equationofseconddegree have real results and their quantity rather than knowing what those results are. through the
discriminating of the quadratic equation, it is possible to know this information.At equationsofseconddegree they can have up to two real and distinct results. In the formula above, note that before the square root there is a “±” sign. This sign only guarantees that one calculation must be done taking the positive value of the result of the root and another calculation must be done taking the negative value of the result of the root. Therefore, up to two results can be found.
Note that if the discriminant is negative, it will not be possible to calculate its root and, therefore, the equation will not have real solutions.
If the discriminant is equal to zero, Bhaskara's formula boils down to:
x = – b ± √Δ
2nd
x = – b ± √0
2nd
x = - B
2nd
As the sign “±” is related to the root, a second degree equation with a discriminant equal to zero will have only one real result.
already the equations with discriminating greater than zero will have two real and distinct results.
So we can say:
If Δ < 0, the equation it has no real results.
If Δ = 0, the equation has a real result.
If Δ > 0, the equation has two real results.
Study of the signs of a function of the second degree
The solution of some problems involving high school functions it can be the range of domain values that causes the counterdomain values to be greater than zero, for example.
It is possible to use the discriminant of equationofseconddegree to determine whether there is a range in which the function is positive or not. For this, keep in mind that the roots of a occupationofsecond degree are its meeting points with the x axis.
If Δ < 0, the function has no roots.
If Δ = 0, the function has a root.
If Δ > 0, the function has two roots.
In addition, the functionsofseconddegree they are parables. Thus, we will have the following possibilities:
If the occupationofseconddegree has Δ > 0, will have two rootsreal and distinct. A part of the parabola that represents it will be above the x-axis and the other below.
If the coefficient a is positive, this function has minimum point below the x-axis, and the occupation it is negative among its roots. otherwise there is peak point above the x-axis, and the function will be positive between its roots.
If the occupationofsecond degree has Δ = 0, will have a real root. So the parable will touch the x-axis at only one point. If a is positive, the entire function is positive except its root (because it is neutral). If a is negative, the entire function will be negative except its root.
If the second degree function has Δ < 0, then it does not have roots. So if a is positive, the entire function will be positive. If a is negative, the entire function will be negative.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-discriminante.htm
