In details

Discriminant


We call it discriminating the radical B2-4ac which is represented by the Greek letter (delta).

We can now write in this way Bhaskara's formula:

According to the discriminant, we have three cases to consider:

1st case: the discriminant is positive .
The value of is real and the equation has two different real roots, represented as follows:

Example:

  • For which values ​​of k the equation x² - 2x + k- 2 = Does it admit real and unequal roots?
    SolutionFor the equation to admit real and unequal roots, we must have

    Therefore, the values ​​of k must be less than 3.


2nd case: the discriminant is null
The value of is null and the equation has two real and equal roots, represented as follows:

Example:

  • Determine the value of P, so that the equation x² - (P - 1) x + p-2 = 0 has equal roots.

    Solution:
    For the equation to admit equal roots, it is necessary that .
    Therefore, the value of P é 3.


3rd case: the discriminant is negative .
The value of doesn't exist in GOtherefore there are no real roots. The roots of the equation are complex number.

Example:

  • For what values ​​of m does equation 3x² + 6x +m = 0 admit no real root?
    Solution:In order for the equation to have no real root, we must have

    Therefore, the values ​​of m must be greater than 3.

summing up

Given the equation ax² + bx + c = 0, we have:

For , the equation has two different real roots.
For , the equation has two equal real roots.
For , the equation has no real roots.

Next: Literal Equations