[Rafael-SOWNet]

Found by a radical-equation cross-check at a seed not used before. No issue was filed.

m_1 sqrt(A) + m_2 sqrt(B) + P = 0, with two different square roots, was not raised at all. It went to Newton's method, which cannot reach a root where a radicand is 0.

	2.6.0	master	this PR
sqrt(-4 - 4x) + sqrt(2 + x) = -3 - 4x	{ -1.19517…, -1 }	{ -1.19517… }	{ -1, … }, the other root in radicals
sqrt(2x + 3) - sqrt(x + 1) = 1	{ -1, 3 }	{ 3 }	{ -1, 3 }
sqrt(-4 - 4x) + sqrt(2 + x) = 1	{ -1 }	unsolved	{ -1 }

The middle column is a regression since 2.6.0, bisected to #2231. 2.6.0 found these roots only because a compiled 0^(-1/2) was 0, so Newton's step from the domain's edge was 0. #2231 made it NaN, which is right, and from then on the root was missed.

The equation is now squared twice, 4 P^2 m_2^2 B = (m_1^2 A - P^2 - m_2^2 B)^2. That keeps every root, and each root of the polynomial is put back into the equation, as for one radical. This happens only for an equation in x alone. With a parameter left in, putting a root back cannot be decided: sqrt(x + 1) + sqrt(x + 2) + a + x = 0 kept all four roots of its square, and at a = 2 none of them is a root.

The same parametric problem exists on master, and in 2.6.0, for one radical: sqrt(x + 1) + a + x = 0 is answered with both roots of its square, and for each a one of them is not a root. That gets its own PR.

radcheck (random radical equations against mpmath), unsolved on master and here:

seed	master	this PR
61	26 / 400	5
51	31 / 400	7
22	24 / 400	3
62, pow	0 / 600	0

No answer is unsound. The one root reported missed, at seed 61, is a harness false positive and is reported against master too: 1 - 1.732i isn't a root of x·sqrt(x² - 2x + 1) = x - 4.

Tests: all eight cases in TwoSquareRootsTest fail on master. There is a BREAKING entry, An equation in two different square roots is solved exactly.

🤖 Generated with Claude Code

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 86.77%. Comparing base (a7785bc) to head (f1af2ee).

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #2335   +/-   ##
=======================================
  Coverage   86.76%   86.77%           
=======================================
  Files         312      312           
  Lines       68526    68540   +14     
  Branches    19935    19938    +3     
=======================================
+ Hits        59458    59473   +15     
  Misses       4907     4907           
+ Partials     4161     4160    -1     

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.31% <100.00%> (+<0.01%)	⬆️
fsharp	32.97% <ø> (ø)
interactive	40.72% <ø> (ø)

Flags with carried forward coverage won't be shown. Click here to find out more.

Files with missing lines	Coverage Δ
...nuous/Solvers/EquationSolver/FractionedPolynoms.cs	85.50% <100.00%> (+3.68%)	⬆️

... and 7 files with indirect coverage changes

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