How do you find the discriminant and how many solutions does #11x^2-9x-1=0?# have?

Question

Nathan Axel · Accepted Answer

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#.

Eva Abramson · Answer

undefined To find the discriminant of a quadratic equation $ ax^2 + bx + c = 0 $, you use the formula: $ \Delta = b^2 - 4ac $. For the equation $ 11x^2 - 9x - 1 = 0 $, where $ a = 11 $, $ b = -9 $, and $ c = -1 $, the discriminant is calculated as follows:

$$ \Delta = (-9)^2 - 4(11)(-1) = 81 + 44 = 125 $$

The discriminant ($ \Delta $) is 125.

Since the discriminant is positive ($ \Delta > 0 $), the quadratic equation $ 11x^2 - 9x - 1 = 0 $ has two distinct real solutions.