Theorems · Theorem · commutative algebra
Polynomial.eq_mul_mul_of_roots_quadratic_eq_pair
∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] {a b c x1 x2 : R},
(Polynomial.C a * Polynomial.X ^ 2 + Polynomial.C b * Polynomial.X + Polynomial.C c).roots = {x1, x2} →
c = a * x1 * x2Vieta's formula for quadratics.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 136 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites35
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- RingHomstatement · cited by 10,189
- Polynomialstatement and proof · cited by 5,681
- mul_oneproof · cited by 3,885
- add_zeroproof · cited by 2,707
- Multisetstatement and proof · cited by 2,627
- zero_addproof · cited by 2,366
- IsDomainstatement and proof · cited by 2,196
- MulZeroClass.mul_zeroproof · cited by 2,091
- le_antisymmproof · cited by 2,068
- mul_assocproof · cited by 1,667
Cited by2
Results whose statement or proof uses this declaration.
- Polynomial.roots_quadratic_eq_pair_iff_of_ne_zeroproof · cited by 2
- Polynomial.eq_mul_mul_of_aroots_quadratic_eq_pairproof · cited by 0