Second order reaction half life equation?
What you did isn't wrong, but instead of:

$$v=-\frac12\frac{d[A]}{dt}=k[A]^2$$

some write:

$$v=-\frac{d[A]}{dt}=k[A]^2$$

which is also true.

Twice a constant is still as constant (a different constant of course).

See http://www.chem.arizona.edu/~salzmanr/480a/480ants/int2ord/int2ord.html