The discriminant is the thing you take the square root of in the

quadratic formula:

#ax^2+bx+c=0# has solution(s): #x=(-b+-sqrt(b^2-4ac))/(2a)#

The discriminant is # b^2-4ac#

If #a,b# and #c# are real numbers, then:

If the discriminant is positive, then there are two real solutions.

#color(white)"ssss"# One when we add and another when we subtract.

If the discriminant is 0, then there is one real solutions.

#color(white)"ssss"# Since #sqrt0 = 0#, adding and subtracting do not give us different answers.

If the discriminant is positive, then there are two imaginary solutions.

#color(white)"ssss"# Since the square root of a negative is imaginary, we get imaginary solutions.

In #11x^2-9x-1=0#, we have

#a=11#, #b=-9# and #c=-1#, so the discriminant is:

# b^2-4ac = (-9)^2-4(11)(-1)#, which is equal to :

#81+44 = 125#

The equation has two real solutions.

#color(white)"ssss"# One when we add #sqrt125# and another when we subtract #sqrt125#.