In the general form of a quadratic expression # f(x) = a x^2 + b x + c #, the discriminant is # Delta = b^2 - 4 a c #. Comparing the given expression with the form, we get # a = -3 #, # b = -4 #, and # c = -3 #. Thus the discriminant is # Delta = (-4)^2 - 4 (-3) (-3) = 16 - 36 = -20 #.
The general solution of the equation # f(x) = 0 # for such a quadratic expression is given by # x = ( -b +- sqrt( Delta )) / (2a) #.
If the discriminant is negative, taking square root would give you imaginary values. In essence, we understand that there are no real solutions of the equation # f(x) = 0 #. This means that the graph of # y = f(x) # never cuts the x-axis. Since # a = -3 < 0 #, the graph is always below the x-axis.
Do note that we do have complex solutions, namely # x = ( -b +- sqrt( Delta )) / (2a) = ( -(-4) +- sqrt( -20 ) ) / (2 (-3) ) = (-4 +- 2sqrt5 i) / (6) = -2/3 +- ( sqrt5 i )/3 #.
