My recommendation is to let e^x become a variable like "u" and convert the equation into a quadratic equation in u, like what you did in the cancelled workings.

The dy/dx path is going on an incorrect track in my opinion since dy/dx = 0 simply tells us to find the turning points.

Proceed on from your cancelled workings, but do not calculate the discriminant at all. The discriminant itself is going on an incorrect track.

3a² + 5a - 2 = 0
(3a + 1) (a - 2) = 0

a = -1/3, a = 2

e^x = -1/3, e^x = 2

We know that any positive number, raised to any real power, will be positive and can therefore not equal -1/3.

We will only have one solution arising from e^x = 2, and that is x = ln 2 = 0.69314... ~ 0.69.