Is there a way for the solve app to use exact figures?

[Albert Chan]

Knowing the limitation of proot and exact, we can use it to get exact results.
But, it required confirmations.

Example: find roots of x^2 + (1+i)*x + (3-i) = 0

XCas> [r1, r2] := proot([1, 1+i, 3-i])
→ [-0.920774266234-2.28242839495*i , -0.0792257337659+1.28242839495*i]

Quadratic roots should have a form of c ± √d = (r1+r2)/2 ± √((r1-r2)/2)²

We use the "inexact" nature of exact, to our advantage.
(If results look messy, we can use the even more "inexact" version, float2rational)

XCas> c := exact((r1+r2)/2)            → -1/(1-i) = -1/2-i/2
XCas> d := exact(((r1-r2)/2)^2)      → (-6+3i)/2 = -3+3i/2

Confimation steps:

XCas> simplify(-2*c)      → 1+i, matching linear coefficient.
XCas> simplify(c*c-d)     → 3-i, matching constant term.

→ x roots = c ± √d = (-1/2-i/2) ± √(-3+3i/2)

For quadratics, this is stupid. We get faster results applying quadratic formula.
But, it is useful for simplifying cubics and beyond.

Example: ³√(1859814842094 - 59687820010√415) = 11589 - 145√415

RHS (actually, just 11589) was originally a guess. Then we confirm it.