We call it **discriminating** the radical **B**** ^{2}-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*x*²*-*2*x + k-*2*=*Does it admit real and unequal roots?**Solution**For 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*x*²*-*(*P -*1*) x + p-2 =*0 has equal roots.For the equation to admit equal roots, it is necessary that .

Solution:

Therefore, the value of*P*é 3.

**3rd case:** the discriminant is negative .

The value of doesn't exist in **GO**therefore there are no real roots. The roots of the equation are **complex number**.

Example:

For what values of m does equation 3

*x*² + 6*x*+*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.

Given the equation ax² + bx + c = 0, we have: For , the equation has two different real roots. |