On the cruelty of really intersecting Bézier curves

To algebraically solve the system $$1+147t-342t^2+221t^3-87u+84u^2-11u^3=0\\ 7-90t+183t^2-104t^3-42u+165u^2-124u^3=0$$ take the resultant with respect to one of the variables of the left sides of the equations, obtaining a polynomial in the other variable, and solve that. The resultant with respect to $u$ is (according to Maple)

$$23266286102592\,{t}^{9}-108607786355952\,{t}^{8}+215982238219020\,{t}^ {7}-238516573272588\,{t}^{6}+159986788335972\,{t}^{5}-67066901150463\, {t}^{4}+17401092642144\,{t}^{3}-2639859118533\,{t}^{2}+202349737962\,t -5453562727 $$

which is irreducible over the rationals. It has five real roots, approximately $$ 0.05361156308, 0.1564617407, 0.4107772234, 0.8657154462, 0.9647259403$$

In the process of finding the resultant by division with remainder, the last step gives us an equation that we can solve for the other variable $u$ in terms of $t$. In this case it turns out to be

$$ \left( {\frac {19344\,{t}^{3}}{47}}-{\frac {1836068\,{t}^{2}}{2867}}+ {\frac {794344\,t}{2867}}-{\frac{541425416}{24659067}} \right) u-{ \frac {21228428\,{t}^{3}}{134749}}+{\frac {2188982416\,{t}^{2}}{ 8219689}}-{\frac {1036030292\,t}{8219689}}+{\frac{3626380}{524661}} $$ so that $$ u = {\frac {31329051\,{t}^{3}-52959252\,{t}^{2}+25065249\,t-1374515}{ 81847116\,{t}^{3}-127355007\,{t}^{2}+55098006\,t-4366334}} $$ Substituting each of the real approximate roots of the resultant into this, we get the approximate $u$ values $$ 0.1008654172, 0.9452315092, 0.8803507865, 0.9713432367, 0.4395155482$$

There's a very good comparison of several intersection algorithms, including most of the ones you mentioned, in this paper by Sederberg and Parry: “Comparison of Three Curve Intersection Algorithms” Computer-Aided Design 18 (1) pp. 58–63, 1986.

Some of the algorithms are based on implicitization. For a good account of that process, and other topics in algebraic geometry, take a look at these notes. Section 17.8 covers curve intersection problems, though it omits much of the detail that's in the paper cited above.