An implicit function $f(x,k)=0$ is quadratic in $x$ and contains one
parameter $k$ which I must vary.
Solve, I get two real solutions for each specific $k$: $x_1$ and $x_2$, of
which I must choose the one that lies within $[0, 0.5]$.
I must do this for a continuum of parameters $k$ between, say, $0$ and $1$.
I then must plot the relevant $x$ solution against the parameter $k$ in a smooth curve.
Note: I also tried the unglorious method i.e. if $k$ took discrete values, I thought I could manually select say 30 points and interpolate. But any curve fitting command I tried, using various polynomial and exponential expansions, could not give me smooth curve?
Of course I hope to learn the elegant method, but under time pressure anything that gives me a smooth curve is welcome!