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PROOF 101

Prove.

The discriminant determines the number of roots

On the real numbers ℝ.

What you may use

A1ax²+bx+c=0 ⟺ (2ax+b)²=D
lemma of Proof 19
A20 < t ⟹ unique s>0 with s²=t
existence of square roots (completeness)
A3u² = v² ⟺ u = ±v
taking square roots
A40 ≤ t²
squares are nonnegative (order)
A5exactly one of a<b, a=b, b<a
trichotomy
A6a < b ⟹ a + c < b + c
addition and order
A7(a+b)+c = a+(b+c)
associativity of addition
A8a + b = b + a
commutativity of addition
A9a + 0 = a
additive identity
A10a + (−a) = 0
additive inverse
A11(ab)c = a(bc)
associativity of multiplication
A12ab = ba
commutativity of multiplication
A13a × 1 = a
multiplicative identity
A14a ≠ 0 ⟹ a·a⁻¹ = 1
multiplicative inverse
A15a(b + c) = ab + ac
distributive law
A16a≠0, b≠0 ⟹ ab ≠ 0
no zero divisors
A170 ≠ 2
2 is not 0
A18t × 0 = 0
lemma of Proof 01
A19there exists i ≠ 0 with i² = −1
imaginary unit (Proof 14)
A20(−a)(−b) = ab
product of negatives (Proof 01)
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

A root of ax² + bx + c = 0 (a ≠ 0) is a number x for which the equation holds upon substitution.

Proof.

Related labs

Equation lab →Quadratic Function lab →