George has a quadratic of the form $x^2+bx+\frac13$, where $b$ is a specific negative number. Using his knowledge of how to complete the square, George is able to rewrite this quadratic in the form $(x+m)^2+\frac{1}{12}$. What is $b$?
x2 + bx + 1/3
= x2 + bx + (b/2)2 - (b/2)2 + 1/3
= (x + b/2)2 - (b/2)2 + 1/3
Now we can see that...
-(b/2)2 + 1/3 = 1/12
-(b/2)2 = 1/12 - 1/3
-(b/2)2 = -1/4
(b/2)2 = 1/4
b2/4 = 1/4
b2 = 1
b = ± 1
b is a negative number, so b = -1