It's been a few years since I've had a problem like this, so hopefully I am going about this correctly. Start by dividing the equation through by 16 to get:
y^2 + y = 1/16
Now, we figure that in order to factor the left hand side of the equation we would need something of the form (y + c)^2 with c denoting some constant value.
We would like to choose something along the lines of:
(y+1/2)^2 because when you multiply it out you get:
y^2 + y + 1/4. But since we have this additional 1/4 on the left side of the equation, we must add it to the right side to get the full equation:
y^2 + y + 1/4 = 1/16 + 1/4
This becomes: (y + 1/2)^2 = (1/4) + (1/16)
which becomes: (y + 1/2)^2 = 5/16.
Now take the square root of both sides to yield:
y + 1/2 = +/- sqrt(5)/4
Final step: subtract 1/2 from both sides to solve for y:
y = (+/- sqrt(5)/4) - 1/2
I hope this explanation is useful.