Quadratic discriminants and roots of a quadratic #

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This file defines the discriminant of a quadratic and gives the solution to a quadratic equation.

Main definition #

discrim a b c: the discriminant of a quadratic a * x * x + b * x + c is b * b - 4 * a * c.

Main statements #

quadratic_eq_zero_iff: roots of a quadratic can be written as (-b + s) / (2 * a) or (-b - s) / (2 * a), where s is a square root of the discriminant.
quadratic_ne_zero_of_discrim_ne_sq: if the discriminant has no square root, then the corresponding quadratic has no root.
discrim_le_zero: if a quadratic is always non-negative, then its discriminant is non-positive.
discrim_le_zero_of_nonpos, discrim_lt_zero, discrim_lt_zero_of_neg: versions of this statement with other inequalities.

Tags #

polynomial, quadratic, discriminant, root

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def discrim {R : Type u_1} [ring R] (a b c : R) :

Discriminant of a quadratic

Equations

discrim a b c = b ^ 2 - 4 * a * c

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@[simp]

theorem discrim_neg {R : Type u_1} [ring R] (a b c : R) :

discrim (-a) (-b) (-c) = discrim a b c

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theorem discrim_eq_sq_of_quadratic_eq_zero {R : Type u_1} [comm_ring R] {a b c x : R} (h : a * x * x + b * x + c = 0) :

discrim a b c = (2 * a * x + b) ^ 2

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theorem quadratic_eq_zero_iff_discrim_eq_sq {R : Type u_1} [comm_ring R] {a b c : R} [ne_zero 2] [no_zero_divisors R] (ha : a ≠ 0) {x : R} :

a * x * x + b * x + c = 0 ↔ discrim a b c = (2 * a * x + b) ^ 2

A quadratic has roots if and only if its discriminant equals some square.

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theorem quadratic_ne_zero_of_discrim_ne_sq {R : Type u_1} [comm_ring R] {a b c : R} (h : ∀ (s : R), discrim a b c ≠ s ^ 2) (x : R) :

a * x * x + b * x + c ≠ 0

A quadratic has no root if its discriminant has no square root.

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theorem quadratic_eq_zero_iff {K : Type u_1} [field K] [ne_zero 2] {a b c : K} (ha : a ≠ 0) {s : K} (h : discrim a b c = s * s) (x : K) :

a * x * x + b * x + c = 0 ↔ x = (-b + s) / (2 * a) ∨ x = (-b - s) / (2 * a)

Roots of a quadratic equation.

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theorem exists_quadratic_eq_zero {K : Type u_1} [field K] [ne_zero 2] {a b c : K} (ha : a ≠ 0) (h : ∃ (s : K), discrim a b c = s * s) :

∃ (x : K), a * x * x + b * x + c = 0

A quadratic has roots if its discriminant has square roots

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theorem quadratic_eq_zero_iff_of_discrim_eq_zero {K : Type u_1} [field K] [ne_zero 2] {a b c : K} (ha : a ≠ 0) (h : discrim a b c = 0) (x : K) :

a * x * x + b * x + c = 0 ↔ x = -b / (2 * a)

Root of a quadratic when its discriminant equals zero

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theorem discrim_le_zero {K : Type u_1} [linear_ordered_field K] {a b c : K} (h : ∀ (x : K), 0 ≤ a * x * x + b * x + c) :

discrim a b c ≤ 0

If a polynomial of degree 2 is always nonnegative, then its discriminant is nonpositive.

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theorem discrim_le_zero_of_nonpos {K : Type u_1} [linear_ordered_field K] {a b c : K} (h : ∀ (x : K), a * x * x + b * x + c ≤ 0) :

discrim a b c ≤ 0

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theorem discrim_lt_zero {K : Type u_1} [linear_ordered_field K] {a b c : K} (ha : a ≠ 0) (h : ∀ (x : K), 0 < a * x * x + b * x + c) :

discrim a b c < 0

If a polynomial of degree 2 is always positive, then its discriminant is negative, at least when the coefficient of the quadratic term is nonzero.

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theorem discrim_lt_zero_of_neg {K : Type u_1} [linear_ordered_field K] {a b c : K} (ha : a ≠ 0) (h : ∀ (x : K), a * x * x + b * x + c < 0) :

discrim a b c < 0