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 .
