What is the discriminant of a quadratic function?
2 Answers
Below
Explanation:
The discriminant of a quadratic function is given by:
What is the purpose of the discriminant?
Well, it is used to determine how many REAL solutions your quadratic function has
If
If
If
Given by the formula
Explanation:
Given a quadratic function in normal form:
#f(x) = ax^2+bx+c#
where
#Delta = b^24ac#
Assuming rational coefficients, the discriminant tells us several things about the zeros of

If
#Delta > 0# is a perfect square then#f(x)# has two distinct rational real zeros. 
If
#Delta > 0# is not a perfect square then#f(x)# has two distinct irrational real zeros. 
If
#Delta = 0# then#f(x)# has a repeated rational real zero (of multiplicity#2# ). 
If
#Delta < 0# then#f(x)# has no real zeros. It has a complex conjugate pair of nonreal zeros.
If the coefficients are real but not rational, the rationality of the zeros cannot be determined from the discriminant, but we still have:

If
#Delta > 0# then#f(x)# has two distinct real zeros. 
If
#Delta = 0# then#f(x)# has a repeated real zero (of multiplicity#2# ).
The discriminant occurs in the quadratic formula for the zeros of
#x = (b+sqrt(b^24ac))/(2a) = (b+sqrt(Delta))/(2a)#
from which you can understand why the zeros have the nature they do for different values of
What about cubics, etc.?
Polynomials of higher degree also have discriminants, which when zero imply the existence of repeated zeros. The sign of the discriminant is less useful, except in the case of cubic polynomials, where it allows us to identify cases quite well...
Given:
#f(x) = ax^3+bx^2+cx+d#
with
The discriminant
#Delta = b^2c^24ac^34b^3d27a^2d^2+18abcd#

If
#Delta > 0# then#f(x)# has three distinct real zeros. 
If
#Delta = 0# then#f(x)# has either one real zero of multiplicity#3# or two distinct real zeros, with one being of multiplicity#2# and the other being of multiplicity#1# . 
If
#Delta < 0# then#f(x)# has one real zero and a complex conjugate pair of nonreal zeros.