Quadratic to Linear solution

Everyone,

I'd love assistance to this apparently simple problem (perhaps not?)

First, we have this usual quadratic equation,
0=a X^2 + b X + c
which will have solutions of
X = -b -(-4a+b^2)^(1/2)/2a , -b +(-4a+b^2)^(1/2)/2a

then we have linear equation,
0= b X + c
with single solution
x = 1/2b

What I have to do is develop somehow a "transition" between these two
extreme limits, allowing the value of "a" go from one to zero, and
somehow have these two equations converge? the "X" is a function, and
I am trying to also make "a" as a function that allows me to explore
the "range" between the limits of quadratic and linear solution
system? Is this even possible? If I solve quadratic equation first,
and do limit with a-> 0, I end up with infinity, instead of linear
solution above.

Any help would be greatly appericated!
.

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