How do the roots of "$x^2 + bx + c$" change as $b$ is kept constant and $c$ is changed? Consider the function $x^2 + bx + c$
How do the (real or complex) roots of the equation change if $b$ is held constant and $c$ is changed? I.e. Which patterns are evident?
What would it look like if you plotted the locus of the roots on an Argand diagram as $c$ is changed?
Also, how do the roots change if $c$ is held constant and $b$ is changed? Again, which patterns (if any) are evident?
 A: (Note: Everything below assumes that $b$ and $c$ are real numbers.)
First, I'll explain how you can get a good handle on the answer this question yourself by hand, and then link to some interactive visualizations. Finally, a bit more work can get a more exact picture.
Exploratory Algebra
The zeros of $x^2+bx+c$ (the roots of $x^{2}+bx+c=0$) are given by $x=\dfrac{-b\pm\sqrt{b^{2}-4c}}{2}$. If the discriminant $b^{2}-4c>0$, then $\sqrt{b^{2}-4c}>0$ and the roots are real and distinct. If $b^{2}-4c=0$, then the roots coincide at $x=-\dfrac{b}{2}$. If $b^{2}-4c<0$, then $\sqrt{b^{2}-4c}=i\sqrt{4c-b^{2}}=i\sqrt{\left|b^{2}-4c\right|}$ and the roots are nonreal and distinct.
For convenience, set $r=-\dfrac{b}{2}$ and $s=\dfrac{\sqrt{\left|b^{2}-4c\right|}}{2}=\sqrt{\left|r^{2}-c\right|}$, and note that since $b$ is kept constant, so is $r$. Then $b^{2}-4c>0$ is equivalent to $c<r^{2}$, and the roots of $x^{2}+bx+c$ can be written as: $x=\begin{cases}r\pm s & \text{ if }c<r^{2}\\r & \text{ if }c=r^{2}\\r\pm is & \text{ if }c>r^{2}\end{cases}$.
Therefore, all the roots lie in a region shaped like $+$ centered at $r$ (the union of the real axis and the vertical line through $r$). Moreover, the distance $s$ from a zero to $r$ is precisely the square root of the absolute difference between $c$ and $r^{2}$, and it's horizontal if $c<r^{2}$ and vertical if $c>r^{2}$.
With calculus, one can derive that as $c$ increases at a constant rate, $s$ is changing at a speed of $\dfrac{1}{2s}$ times that rate (so the roots are moving arbitrarily fast near $r$).

Interactives
Here is an interactive I just made in Desmos where you can watch the locations of the roots move as you vary $b$ and/or $c$. If you already have the Wolfram CDF player or have access to Mathematica, you can play with a similar interactive previously developed by Robert Baillie here.

More Algebra
Instead of an interactive/animation, we can get a clearer picture by figuring out exactly what the picture of the dependence of the roots on $c$ is. In other words, by preparing a 3D picture of the real and complex parts of the roots for each $c$. We can do this with a computer just from the work above, but we wouldn't be certain of the exact shape of the picture. An approach that will work is to create a pair of parametric functions on the axes $\text{real part of }x,\text{imaginary part of }x,c$. Since $x^2+bx+c=0$, $c=-x^2-bx$, so that $c$ can be written as a function of a root.
When the pair of roots is real, they're just $r\pm s$, so we can use something in place of $s$ as a parameter ($c$ can be written as a function of $s$). Then $\mathbf{f}(t)=\left(t+r,0,-(t+r)^2-b(t+r)\right)=\left(t+r,0,-(t+r)^2+2r(t+r)\right)$ is a parametric plot of the locations of the real roots, with $t=0$ corresponding to the double-root. Note that this is a parabola in our 3D space.
When the pair of roots is nonreal, they are $r\pm is$, so the real part is $r$ and $-x^2-bx=-(r\pm is)^2+2r(r\pm is)=r^{2}+s^{2}$. Then $\mathbf{g}(t)=\left(r,t,t^2+r^2\right)$ is a parametric plot of the locations of the complex roots, with $t=0$ corresponding to the double-root. Note that this is another parabola in our 3D space.

Static 3D picture
Plotting the $\mathbf{f}$ and $\mathbf{g}$ in the same space, we obtain a picture like this (here $r=1$):

The Mathematica code I used is
Show[ParametricPlot3D[{{x + 1, 0, 2 (x + 1) - (x + 1)^2}, {1, x, 1 + x^2}}, {x, -2, 2}, 
BoxRatios -> Automatic, PlotStyle -> Green, Boxed -> False, AxesOrigin -> {0, 0, 0}, AxesStyle -> Directive[Thick]] 
/. Line[pts_] :> Tube[pts, .05], 
Graphics3D[{Text["Real", {3.5, 0, 0}], Text["Imag", {0, 2.5, 0}], Text[C, {0, 0, 5.5}]}]]
