# Unexpected extra result when I don't simplify before solving

Why am I seeing two different results below?

Solve[2 t^2 - 4.6 t + 1.2 - (1 - 4 t) (0.3 t - 1) == 3.2 t^2]

{{t -> -4.0082*10^16}, {t -> 0.247191}}

Solve[FullSimplify[2 t^2 - 4.6 t + 1.2 - (1 - 4 t) (0.3 t - 1)] == 3.2 t^2]

{t -> 0.247191}}


Up to machine precision, the "wrong" solution is indeed correct:

a = 2 t^2 - 4.6 t + 1.2 - (1 - 4 t) (0.3 t - 1) /.
t -> -4.008203668359741*^16;
b = 3.2 t^2 /. t -> -4.008203668359741*^16;
a == b
(a - b)/a < \$MachineEpsilon


True

True

Taking the differences of both sides of the equation also shows us, why this happens:

2 t^2 - 4.6 t + 1.2 - (1 - 4 t) (0.3 t - 1) - 3.2 t^2 // Expand


2.2 - 8.9 t - 2.22045*10^-16 t^2

So, this is really on the edge between quadratic and linear equations.

This tells us that we have to be cautious when solving in machine precision. Often, applying Rationalize to the equations before solving does also help.