Apparently, I got this question wrong on a test and I'd like to know what error I made... can you help?

Question

Here's what I did:

2b^2+b=6
b^2+1/2b=3
b^2+1/2b+(1/2)^2=3+(1/2)^2
(b+1/2)^2=3+1/4
b+1/2=+/- <13/4>
b=-1/2+/-<13>/2
b=(-1+/-<13>)/2

<> means its in a radical and +/- means plus or minus

Answer 1

The error is in your third line

you take 1/2 of the middle coefficient, and then square it
(1/2)(1/2) = 1/4 --- (1/4)^2 = 1/16

3rd line:

b^2 + (1/2)b + 1/16 = 3 + 1/16
(b+1/4)^2 = 49/16
b + 1/4 = ± 7/4
b = -1/4 ± 7/4 = 6/4 or -8/4
b = -2 or 3/2

since we have rational answers, we know it would have factored to

(x+2)(2b-3)
= 2b^2 + b - 6 , the original.