The examples in this subsection have been used for testing purposes by Richard Lang. The equations contain roots, which as usual are eliminated by squaring the equations; this, however, generates assumptions, which presently are not handled nicely (ISAC still lacks a simplifier for root-expressions). Thus this subsection includes some equations, which end up in incorrect results.

Solve the root-equations in x ...

No.1 4 * sqrt(4*x + 1)=3 * sqrt(7*x + 2)

One root on a side of the equation leads to a linear equation.
No.2 sqrt(x + 1) + sqrt(4*x + 4)=sqrt(9*x + 9)

Two roots on one side of the equation usually leads to a equation of degree two. However, this particular equation turns out to hold for any x.
No.3 sqrt(4*x + 1) - sqrt(x + 3) = sqrt(x - 2)

This equation reduces to a linear equation, luckly.
No.4 sqrt(x + 12) + sqrt(x - 3) = sqrt(x + 32) - sqrt(5 + x)

This equation is calculated correcty, but finally fails due to the lack of a canonical simplifier for formulae with roots -- thus a nice todo.
No.5 sqrt(29 - sqrt(x^2 - 9)) = 5

This equation has two solutions.
No.6 2*sqrt(261 - x) - sqrt(2 + 2*x) = sqrt(2)*sqrt(5 - 3*x)

There is something wrong --- but you can trace, how it goes wrong !
No.7 (2*x + 1) * x^2 = 0

This one has three solutions, as expected (one of them a 'double' solution).
No.8 b^2 * x^2 + a^2 * y^2 = a^2 * b^2

Finally there are some root-equations with another bound variable than x.

No.9 (a/2)^2 + (b/2)^2 = r^2 ,  a = ?
No.10 2*sqrt(r^2 - (u/2)^2) - u^2/(2*sqrt(r^2 - (u/2)^2)) = 0 ,  u = ?
No.11 y^2 = 2*p*x ,  y = ?