Q:

WolframAlpha's problems with equations involving the floor operation

The whole story began when I was developing an easy way to solve complex equations involving mix-up of absolute value, floor, ceiling, rounding, and other functions, as well as polynomials embedded in them.
When I was pretty much done, I started solving different equations like that. I came up with couple equations. One of them is:
$$|x^2 + 3x| + \lfloor2x^2\rfloor + x = 5$$
I ended up with eight numbers. I expected six of them to be extraneous, and two of them to work. However, it turned out, none of the eight numbers worked. So I concluded there are no solutions. Just to be sure, I opened up Wolfram Alpha and wanted to see if there is a solution: http://www.wolframalpha.com/input/?i=abs(x%5E2+%2B+3x)+%2B+floor(2x%5E2)+%2B+x+%3D+5
Wolfram Alpha spitted out $x=2\sqrt{2} - 2$.
I checked the solution that WA gave me, but to my surprise it didn't work! If I plugged it inside wolfram alpha for $x$, it worked, but when I did it on paper, it didn't!
Out of curiosity, I typed in this equation in Wolfram Alpha:
$$\lfloor2x^2\rfloor = x+2$$
And, now I was even more surprised. WA gave me solution $-1$, which IS total NONSENSE. It can be checked here: http://www.wolframalpha.com/input/?i=floor(2x%5E2)+%3D+x%2B2 . Obviously $-1$ doesn't work in that equation.
Most important problem: So, I know that Wolfram Alpha gave me the wrong answer for the second equation, however I am still not sure about the first one. Is the solution WA gave for the first equation correct, or am I right about that there is no solution?
ANSWER
I was indeed "hallucinating". One of the eight numbers I got was actually the solution Wolfram Alpha gave me, although I got it in a different form, and somehow it didn't work when I tested it.
Still, the bug present in the second equation needs to be fixed, I'll report it.

A:

What's happening (for the second one) is that, for example
$$ \lfloor 2(-0.99)^2 \rfloor - (-0.99) = \lfloor 1.9602 \rfloor + 0.99 = 1.99 \approx 2. $$
So what WolframAlpha does, when trying to solve numerically, is get $-0.9999\dots$ (to some chosen precision) which is very close as long as there are only a finite number of '9's. It then rounds (when printing) and gets $-1$ which is no longer a solution.

The first one is a solution:
\begin{align}
&\qquad |(2\sqrt 2 - 2)^2 + 3(2\sqrt 2 - 2)| + \lfloor 2(2\sqrt 2 - 2)^2 \rfloor + (2\sqrt 2 - 2) \\
&= |6 - 2\sqrt 2| + \lfloor 1.37 \rfloor + (2\sqrt 2 - 2)  \\
&= 5.
\end{align}

