00.03.02 · precalc / equations-lines

Quadratic equations and the quadratic formula

shipped3 tiersLean: partial

Anchor (Master): al-Khwārizmī ~825 Al-jabr; Cardano 1545 Ars Magna; Viète 1591 In artem analyticen isagoge; Gauss 1801 Disquisitiones Arithmeticae; Lang Algebra Ch. IV §1

Intuition [Beginner]

A quadratic equation in one variable $x$ has the form $a x^{2} + b x + c = 0$ , where $a$ , $b$ , $c$ are numbers and the leading coefficient $a$ is not zero. The number we square is what makes it quadratic, and the condition $a \neq = 0$ is what prevents the equation from collapsing into a linear one. The quadratic formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

gives the solutions directly from the coefficients. The expression $b^{2} - 4 a c$ inside the square root is called the discriminant, written $Δ$ . It controls the answer to the central question: how many real solutions are there?

The discriminant tells us three things at once. If $Δ > 0$ the square root is a positive real number, so the two values $(- b + Δ) / (2 a)$ and $(- b - Δ) / (2 a)$ are different — two distinct real solutions. If $Δ = 0$ the square root is zero, the $\pm$ collapses, and there is one repeated real solution $- b / (2 a)$ . If $Δ < 0$ the number under the square root is negative; no real number squares to a negative, so the equation has no real solutions — only two complex ones.

The geometric picture matches the algebra. The graph of $y = a x^{2} + b x + c$ is a parabola — a U-shaped curve opening upward when $a > 0$ and downward when $a < 0$ . The solutions of $a x^{2} + b x + c = 0$ are exactly the $x$ -values where the parabola crosses the $x$ -axis. A parabola that crosses the axis twice has two real roots, one that just touches the axis at its vertex has one repeated root, and one that floats entirely above or below the axis has no real roots.

Visual [Beginner]

Picture a parabola drawn on a coordinate grid, opening upward and crossing the $x$ -axis at two marked points. The curve dips below the axis between the crossings and rises above the axis on both sides. The vertex of the parabola sits at the bottom of the dip, marked with a smaller dot. The two crossing points are the solutions of the quadratic equation that defines the curve.

The picture records the central correspondence. Each real root of $a x^{2} + b x + c = 0$ is a crossing of the parabola $y = a x^{2} + b x + c$ with the horizontal axis. The number of real roots — two, one, or zero — matches the number of crossings. The discriminant $b^{2} - 4 a c$ measures, algebraically, how far the parabola sits from being tangent to the axis: positive means cleanly two crossings, zero means tangent at the vertex, and negative means the parabola misses the axis entirely.

Worked example [Beginner]

Solve $x^{2} - 5 x + 6 = 0$ . Read off the coefficients: $a = 1$ , $b = - 5$ , $c = 6$ . Compute the discriminant: $Δ = b^{2} - 4 a c = (- 5)^{2} - 4 \cdot 1 \cdot 6 = 25 - 24 = 1$ . Since $Δ > 0$ there are two distinct real solutions. The square root of the discriminant is $1 = 1$ . Apply the formula: $x = (5 \pm 1) /2$ , giving $x = 3$ or $x = 2$ .

Check by factoring: $(x - 2) (x - 3) = x^{2} - 5 x + 6$ , so the equation is $(x - 2) (x - 3) = 0$ , with solutions $x = 2$ and $x = 3$ . The two answers match.

Solve $x^{2} + 1 = 0$ . Here $a = 1$ , $b = 0$ , $c = 1$ , so $Δ = 0 - 4 = - 4 < 0$ . The discriminant is negative, so the equation has no real solutions. The parabola $y = x^{2} + 1$ sits one unit above the $x$ -axis at its vertex and rises from there, never crossing the axis. Over the complex numbers the formula gives $x = (\pm - 4) /2 = \pm i$ , the two imaginary solutions.

What this tells us: the discriminant predicts the answer before any computation. A positive discriminant gives two real roots, a zero discriminant gives a repeated real root, and a negative discriminant gives no real roots. The quadratic formula then turns those predictions into specific numbers.

Check your understanding [Beginner]

Formal definition [Intermediate+]

A quadratic polynomial over a field $k$ is a polynomial $p (x) = a x^{2} + b x + c \in k [x]$ with $a, b, c \in k$ and $a \neq = 0$ , the leading coefficient. A quadratic equation is the equation $p (x) = 0$ for a quadratic polynomial $p$ ^{[Lang — Basic Mathematics Ch. 3–4]}. A root (or solution) of the equation is an element $r \in k$ (or in an extension field) with $p (r) = 0$ .

The discriminant of the quadratic polynomial $p (x) = a x^{2} + b x + c$ is

Δ = b^{2} - 4 a c \in k .

The vertex of the parabola $y = a x^{2} + b x + c$ , viewed as a curve in $k^{2}$ , is the point $(x_{v}, y_{v})$ with $x_{v} = - b / (2 a)$ and $y_{v} = c - b^{2} / (4 a) = - Δ/ (4 a)$ . This is the unique critical point of the polynomial $p$ as a function of $x$ — the place where the derivative $p^{'} (x) = 2 a x + b$ vanishes — and it is the global minimum when $a > 0$ and global maximum when $a < 0$ .

The quadratic polynomial $p$ admits the completed-square form

p (x) = a (x + \frac{b}{2 a})^{2} - \frac{Δ}{4 a},

a pointwise identity in $k [x]$ valid whenever $2 \in k$ is invertible (so over any field of characteristic not two). The completed-square form makes the vertex coordinates visible: the polynomial equals $- Δ/ (4 a)$ plus a non-negative multiple of $a$ at every $x$ , reaching the value $- Δ/ (4 a)$ exactly when $x = - b / (2 a)$ .

Counterexamples to common slips

The condition $a \neq = 0$ is essential. If $a = 0$ the polynomial $p (x) = b x + c$ is linear, not quadratic, and the quadratic formula's denominator $2 a$ vanishes. The classification by discriminant does not extend to the linear case in a meaningful way.
The discriminant depends on the polynomial, not just its set of roots. The polynomials $p (x) = x^{2} - 5 x + 6$ and $q (x) = 2 x^{2} - 10 x + 12 = 2 p (x)$ have the same roots ${2, 3}$ , but their discriminants are $1$ and $4$ , respectively. Multiplying the equation by a nonzero constant scales the discriminant by the square of that constant.
Over a field of characteristic two the formula breaks down: division by $2 a = 0$ is undefined. Quadratic equations over $F_{2}$ and its extensions require a different treatment via Artin-Schreier theory rather than the quadratic formula.

Key theorem with proof [Intermediate+]

Theorem (the quadratic formula via completing the square). Let $k$ be a field of characteristic not two and let $a, b, c \in k$ with $a \neq = 0$ . Set $Δ = b^{2} - 4 a c$ . Then the equation $a x^{2} + b x + c = 0$ has the solutions

x = \frac{- b \pm Δ}{2 a}

in any extension field $K \supseteq k$ in which $Δ$ admits a square root ^{[Lang — Basic Mathematics Ch. 3–4]}. When $Δ$ has a square root in $k$ itself, the two solutions lie in $k$ ; when $Δ = 0$ the two solutions coincide as the single repeated root $- b / (2 a)$ ; and when $Δ$ is not a square in $k$ , the two solutions lie in the quadratic extension $k (Δ)$ .

Proof. Since $a \neq = 0$ and $k$ has characteristic not two, both $a$ and $2 a$ are invertible. Divide the equation by $a$ :

x^{2} + \frac{b}{a} x + \frac{c}{a} = 0.

Add and subtract $(b / (2 a))^{2}$ on the left:

(x^{2} + \frac{b}{a} x + \frac{b ^{2}}{4 a ^{2}}) - \frac{b ^{2}}{4 a ^{2}} + \frac{c}{a} = 0.

The first three terms form a square: $x^{2} + (b / a) x + b^{2} / (4 a^{2}) = (x + b / (2 a))^{2}$ . Substitute and rearrange:

(x + \frac{b}{2 a})^{2} = \frac{b ^{2}}{4 a ^{2}} - \frac{c}{a} = \frac{b ^{2} - 4 a c}{4 a ^{2}} = \frac{Δ}{4 a ^{2}} .

Take square roots in any extension field $K$ in which $Δ$ has a square root $s$ :

x + \frac{b}{2 a} = \pm \frac{s}{2 a},

using that $(2 a)^{2} = 4 a^{2}$ and $2 a \neq = 0$ . Solve for $x$ :

x = \frac{- b \pm s}{2 a} = \frac{- b \pm Δ}{2 a} .

Substituting either value back into $a x^{2} + b x + c$ and using the completed-square identity confirms the value is zero. The two solutions coincide iff $s = 0$ iff $Δ = 0$ . $□$

Corollary (Vieta's formulas). If $x_{1}$ and $x_{2}$ are the two roots of the monic quadratic $x^{2} + p x + q = 0$ over a field, counted with multiplicity, then $x_{1} + x_{2} = - p$ and $x_{1} x_{2} = q$ . For the general quadratic $a x^{2} + b x + c = 0$ the corresponding identities are $x_{1} + x_{2} = - b / a$ and $x_{1} x_{2} = c / a$ .

Proof of corollary. Multiply out $(x - x_{1}) (x - x_{2}) = x^{2} - (x_{1} + x_{2}) x + x_{1} x_{2}$ and compare coefficients with $x^{2} + p x + q$ , or compute directly from $x_{1, 2} = (- b \pm Δ) / (2 a)$ : the sum is $(- b + Δ + (- b) - Δ) / (2 a) = - b / a$ , and the product is $((- b)^{2} - Δ) / (4 a^{2}) = (b^{2} - (b^{2} - 4 a c)) / (4 a^{2}) = c / a$ .

Bridge. The quadratic formula builds toward the broader structure theory of polynomial equations and the symmetry theory of roots. First, completing the square is the foundational substitution technique: by setting $y = x + b / (2 a)$ one eliminates the linear term, reducing the equation to $y^{2} = Δ/ (4 a^{2})$ . The analogous depression substitution for the cubic $x^{3} + p x^{2} + q x + r$ removes the quadratic term by setting $y = x + p /3$ , reducing the equation to a depressed cubic $y^{3} + P y + Q = 0$ amenable to Cardano's 1545 formula; for the quartic, Ferrari's 1540 method depresses to $y^{4} + P y^{2} + Q y + R$ and reduces via a resolvent cubic. Second, the discriminant $Δ = b^{2} - 4 a c$ generalises: every polynomial $p (x)$ of degree $n$ over a field has a discriminant $Δ_{n} = a_{n}^{2 n - 2} \prod_{i < j} (r_{i} - r_{j})^{2}$ , a polynomial in the coefficients (via Vieta) that vanishes precisely when $p$ has a repeated root. The discriminant of a polynomial detects ramification in algebraic number theory and the singular locus in algebraic geometry. Third, Vieta's identities $x_{1} + x_{2} = - b / a$ and $x_{1} x_{2} = c / a$ are the prototype of the general fact that the elementary symmetric functions of the roots of a monic polynomial $\prod (x - r_{i})$ are (up to sign) the coefficients — the symmetric-function theory worked out by Newton and Lagrange and used systematically by Galois.

Putting these together, the foundational insight is that the formula for the roots of a polynomial is intertwined with the symmetry theory of the roots. The Abel-Ruffini theorem (1824) — that no general formula in radicals exists for polynomials of degree five or higher — is the negative companion of the formulas of Cardano and Ferrari. Galois theory (1832) supplied the structural explanation: the solvability of a polynomial equation in radicals is governed by the solvability of the associated Galois group, the group of permutations of the roots that preserves the rational relations among them. The quadratic and cubic and quartic each have soluble Galois groups (subgroups of $S_{n}$ for $n \leq 4$ ), while the generic quintic has Galois group $S_{5}$ , which is not soluble. The quadratic formula is the simplest entry point to a body of algebra that runs from al-Khwārizmī's geometric construction in the ninth century to the structure-theoretic Galois correspondence of the nineteenth, and forward into modern algebraic number theory and algebraic geometry through the discriminant.

Exercises [Intermediate+]

Exercise 6 (hard, symbolic).

Show that for any $a, b, c \in R$ with $a \neq = 0$ , the parabola $y = a x^{2} + b x + c$ is tangent to the $x$ -axis (touches it at exactly one point without crossing) if and only if the discriminant $Δ = b^{2} - 4 a c$ equals zero. Identify the point of tangency in that case.

Hint

A line is tangent to the parabola at a point where the parabola meets the line and the parabola's derivative matches the line's slope. The $x$ -axis has slope zero.

Answer

The parabola meets the $x$ -axis at the real roots of $a x^{2} + b x + c = 0$ . By the quadratic formula, the equation has exactly one (repeated) real root iff $Δ = 0$ , the two roots from the formula merging into the single value $x_{v} = - b / (2 a)$ . At this point the parabola's derivative is $p^{'} (x_{v}) = 2 a x_{v} + b = - b + b = 0$ , matching the slope of the $x$ -axis, so the parabola is tangent to the axis at $x_{v}$ . Conversely, if $Δ > 0$ the parabola crosses the axis at two distinct points (transverse intersections, not tangencies), and if $Δ < 0$ the parabola does not meet the axis at all. Hence tangency holds iff $Δ = 0$ , and the point of tangency is the vertex $(x_{v}, 0) = (- b / (2 a), 0)$ .

Rubric: full credit for identifying the tangency condition with $Δ = 0$ , computing the tangency point as the vertex, and addressing both directions of the equivalence (positive and negative discriminant cases).

Exercise 7 (hard, short-answer).

A quadratic polynomial $p (x) = a x^{2} + b x + c$ over the real numbers has its vertex at $(h, k)$ and passes through the point $(h + 1, k + a)$ . Show that $p$ has the form $p (x) = a (x - h)^{2} + k$ , and express its discriminant in terms of $a$ , $h$ , and $k$ .

Hint

The vertex-form of a parabola exposes the vertex directly. Verify the given point and convert back to standard form to read off the discriminant.

Answer

The general vertex-form of a parabola with leading coefficient $a$ and vertex $(h, k)$ is $p (x) = a (x - h)^{2} + k$ . At $x = h + 1$ : $p (h + 1) = a (1)^{2} + k = a + k$ , matching the given point $(h + 1, k + a)$ . Expanding to standard form: $p (x) = a x^{2} - 2 ah x + (a h^{2} + k)$ , so $b = - 2 ah$ and $c = a h^{2} + k$ . The discriminant is $Δ = b^{2} - 4 a c = 4 a^{2} h^{2} - 4 a (a h^{2} + k) = 4 a^{2} h^{2} - 4 a^{2} h^{2} - 4 ak = - 4 ak$ . So $Δ = - 4 ak$ , expressing the discriminant entirely in terms of the leading coefficient and the vertex height. Sanity check: the parabola crosses the $x$ -axis ( $Δ > 0$ ) iff $a$ and $k$ have opposite signs — exactly when the vertex sits on the opposite side of the axis from the opening direction.

Rubric: full credit for verifying the vertex form, expanding back to standard form to find $b$ and $c$ , and arriving at $Δ = - 4 ak$ with a sanity check.

Lean formalization [Intermediate+]

[object Promise]

The named statements compile against Mathlib's Algebra.QuadraticDiscriminant. Mathlib supplies the discriminant under the name discrim a b c = b^2 - 4ac, the characterisation quadratic_eq_zero_iff giving the root pair when a square root of the discriminant exists, the trichotomy discrim_lt_zero / discrim_eq_zero / discrim_pos for real-coefficient quadratics, and Vieta-style identities for the sum and product of roots. The human reviewer named in the frontmatter signs off on the coverage claim.

Advanced results [Master]

The quadratic formula is the simplest entry point to the structural theory of polynomial equations, and the generalisations follow in coordinated directions: higher degree, multiple variables, finite fields, and number-theoretic reciprocity.

Cubic and quartic formulas. The depression substitution $y = x + b / (3 a)$ transforms $a x^{3} + b x^{2} + c x + d = 0$ into the depressed cubic $y^{3} + p y + q = 0$ with $p = (3 a c - b^{2}) / (3 a^{2})$ and $q = (2 b^{3} - 9 ab c + 27 a^{2} d) / (27 a^{3})$ . Cardano's formula expresses the roots via nested radicals: setting $y = u + v$ with $3 uv = - p$ and $u^{3} + v^{3} = - q$ reduces the cubic to a quadratic in $u^{3}$ with discriminant $Δ_{Cardano} = q^{2} + 4 p^{3} /27$ ^{[Cardano — Ars Magna]}. Ferrari's quartic formula, published in the same 1545 volume, depresses $a x^{4} + \dots$ to $y^{4} + P y^{2} + Q y + R = 0$ via $y = x + b / (4 a)$ and then introduces an auxiliary parameter $z$ to write the depressed quartic as a difference of two squares, reducing to a resolvent cubic. Both formulas exhibit the same pattern as the quadratic formula: a substitution removes the second-highest-degree term, and the reduced equation is solved by an explicit radical expression in the coefficients.

Abel-Ruffini and the Galois obstruction. No analogous closed-form radical solution exists for the general polynomial of degree five or higher. Paolo Ruffini's 1799 Teoria generale delle equazioni gave an incomplete proof, completed by Niels Henrik Abel in 1824: there is no formula in radicals — finite expressions involving the coefficients, the field operations, and $n$ -th roots — that solves the general quintic ^{[Lang — Algebra Ch. VI]}. Évariste Galois's 1832 manuscript supplied the structural explanation. To each polynomial $f \in k [x]$ one associates its Galois group $Gal (K / k)$ , the group of automorphisms of a splitting field $K$ fixing $k$ pointwise. A polynomial is solvable by radicals iff its Galois group is soluble — that is, admits a composition series with abelian quotients. The Galois groups of generic quadratics, cubics, and quartics are subgroups of $S_{2}, S_{3}, S_{4}$ (all soluble), while the generic quintic has Galois group $S_{5}$ , which contains the simple non-abelian subgroup $A_{5}$ and is not soluble. The quadratic case is the apex of solubility: $S_{2} = Z /2$ is abelian, the discriminant generates the splitting field over the base field as $k (Δ)$ , and the formula's single square root is the simplest possible radical.

Discriminant of a polynomial. For a polynomial $f (x) = a_{n} \prod_{i = 1}^{n} (x - r_{i})$ of degree $n$ over a field, the discriminant is

Δ (f) = a_{n}^{2 n - 2} i < j \prod (r_{i} - r_{j})^{2} .

The product over $i < j$ is symmetric in the roots, hence — by the fundamental theorem of symmetric functions — expressible as a polynomial in the coefficients of $f$ . For the quadratic $f (x) = a x^{2} + b x + c$ this recovers $Δ_{2} = a^{2 \cdot 2 - 2} (r_{1} - r_{2})^{2} = a^{2} \cdot ((r_{1} + r_{2})^{2} - 4 r_{1} r_{2}) = a^{2} \cdot (b^{2} / a^{2} - 4 c / a) = b^{2} - 4 a c$ . The discriminant vanishes iff $f$ has a repeated root, which over a perfect field is iff $f$ shares a common factor with its derivative $f^{'}$ . In algebraic number theory, the discriminant of the minimal polynomial of an algebraic number generates the discriminant ideal of the corresponding ring of integers and detects ramified primes; in algebraic geometry, the discriminant locus is the singular locus of the family of zero sets parametrised by the coefficients.

Quadratic forms and Sylvester's law of inertia. A quadratic form on $R^{n}$ is a homogeneous polynomial of degree two, equivalently $Q (v) = v^{⊤} A v$ for a symmetric matrix $A \in Sym_{n} (R)$ . The spectral theorem gives an orthogonal change of basis $v = P y$ with $P^{⊤} P = I$ and $P^{⊤} A P = diag (λ_{1}, \dots, λ_{n})$ , the $λ_{i}$ the eigenvalues of $A$ . In the new coordinates the form reads $Q = \sum_{i} λ_{i} y_{i}^{2}$ . The number $p$ of positive eigenvalues, the number $q$ of negative eigenvalues, and the number $n - p - q$ of zero eigenvalues are invariants of $Q$ under arbitrary change of basis — not just orthogonal — by Sylvester's law of inertia (Sylvester 1852). The pair $(p, q)$ is the signature of the form, and the signature is a complete invariant: two quadratic forms over $R$ are equivalent (related by a basis change) iff they have the same signature. The one-variable case $Q (x) = a x^{2}$ has signature $(1, 0)$ for $a > 0$ , $(0, 1)$ for $a < 0$ , and $(0, 0)$ for $a = 0$ — recovering the sign of the leading coefficient of a quadratic polynomial as the simplest signature invariant.

Conics and the binary discriminant. A conic section in the affine plane $R^{2}$ is the zero set of a degree-two polynomial $C (x, y) = a x^{2} + b x y + c y^{2} + d x + ey + f$ . The conic discriminant is $Δ_{C} = b^{2} - 4 a c$ , computed from the homogeneous degree-two part alone. The sign of $Δ_{C}$ classifies the conic up to affine equivalence: $Δ_{C} < 0$ gives an *ellipse* (the matrix of the quadratic part is sign-definite), $Δ_{C} = 0$ a *parabola* (degenerate, the quadratic part factors as a square of a linear form), and $Δ_{C} > 0$ a hyperbola (sign-indefinite quadratic part) ^{[Apostol — Calculus Vol. 1 §I.4]}. The classification reduces to the one-variable discriminant in this way: the homogeneous degree-two part $a x^{2} + b x y + c y^{2}$ viewed as a quadratic polynomial in $x$ with $y$ a parameter has discriminant $(b y)^{2} - 4 a c y^{2} = (b^{2} - 4 a c) y^{2} = Δ_{C} y^{2}$ , and the sign of $Δ_{C}$ determines whether the conic decomposes into two real lines through the origin (the asymptotic cone of the conic, hyperbolic case), one repeated real line (parabolic case), or no real lines (elliptic case).

Quadratic reciprocity. For distinct odd primes $p$ and $q$ , the Legendre symbol $(\frac{p}{q})$ is $+ 1$ if $p$ is a quadratic residue modulo $q$ (the equation $x^{2} \equiv p (mod q)$ has a solution), $- 1$ otherwise, and $0$ if $q ∣ p$ . The law of quadratic reciprocity, proved by Gauss in his 1801 Disquisitiones Arithmeticae and called by him the aureum theorema (golden theorem), is the identity

(\frac{p}{q}) (\frac{q}{p}) = (- 1)^{(p - 1) (q - 1) /4}

^{[Gauss — Disquisitiones Arithmeticae]}. Together with the supplementary laws for $(\frac{- 1}{p})$ and $(\frac{2}{p})$ , reciprocity reduces the question of whether the quadratic equation $x^{2} \equiv a (mod p)$ has a solution to a finite computation in the residues modulo small primes. Gauss gave eight different proofs in his lifetime; the result is the deepest theorem of elementary number theory and the foundational case of the broader class-field-theoretic reciprocity laws (Artin reciprocity, the Langlands program).

Algebraic closure. Over the complex numbers $C$ , every quadratic equation $a x^{2} + b x + c = 0$ with $a \neq = 0$ has two solutions counted with multiplicity, since every element of $C$ has a square root (in fact $n$ distinct $n$ -th roots for any $n \geq 1$ ). The fundamental theorem of algebra — first stated by d'Alembert in 1746, with the first essentially complete proof in Gauss's 1799 doctoral dissertation Demonstratio nova theorematis — extends this from quadratics to all polynomials: every non-constant $p \in C [x]$ has at least one root in $C$ , hence factors completely into linear factors. The closure of $R$ under taking quadratic roots — adjoining $- 1 = i$ to form $C = R [i] = R [x] / (x^{2} + 1)$ — turns out to be the closure under taking roots of every polynomial, a fact special to the real numbers among archimedean fields.

Synthesis. The bridge between the elementary quadratic formula and the broader theory of polynomial equations is the recognition that the discriminant $Δ = b^{2} - 4 a c$ is the load-bearing invariant. Once one has the discriminant, the trichotomy of solution counts, the conic classification, the binary-quadratic-form signature, the discriminant of an algebraic number field, and the splitting-field structure $k (Δ)$ all read off as instances of the same algebraic shadow. The quadratic generalises in degree (cubic, quartic, quintic), in field (rational, real, complex, finite, $p$ -adic), in variable count (binary quadratic form, ternary quadratic form, $n$ -ary), and in arithmetic content (Legendre symbols, quadratic reciprocity, class-field-theoretic reciprocity). Completing the square generalises to the depression substitutions of Cardano and Ferrari, to the diagonalisation of quadratic forms by orthogonal change of basis, and to the normal-form reductions throughout linear algebra. Vieta's formulas generalise to the elementary symmetric functions of the roots of a degree- $n$ polynomial, the foundation of Newton's identities, Lagrange's resolvents, and Galois's symmetry-group construction.

This unit identifies the quadratic polynomial as the prototype higher-degree polynomial, the discriminant as the prototype algebraic invariant detecting repeated roots, completing the square as the prototype substitution that diagonalises a polynomial, the quadratic formula as the prototype radical expression of the roots in terms of the coefficients, and Vieta's identities as the prototype symmetric-function expression of the coefficients in terms of the roots. Each of these prototype roles motivates a downstream generalisation that runs through the algebra and geometry strands of the curriculum.

Full proof set [Master]

Proposition (real-discriminant trichotomy). Let $a, b, c \in R$ with $a \neq = 0$ and let $Δ = b^{2} - 4 a c$ . Then the equation $a x^{2} + b x + c = 0$ has two distinct real roots iff $Δ > 0$ , a single repeated real root iff $Δ = 0$ , and no real roots (two complex-conjugate roots) iff $Δ < 0$ .

Proof. The quadratic formula gives $x = (- b \pm Δ) / (2 a)$ over $C$ . When $Δ > 0$ the square root $Δ$ is a positive real, so the two values are real and distinct. When $Δ = 0$ the two values both equal $- b / (2 a) \in R$ , a single repeated real root. When $Δ < 0$ the square root is $Δ = i - Δ$ with $- Δ > 0$ , so $x = (- b \pm i - Δ) / (2 a)$ is a pair of complex-conjugate non-real numbers, and there are no real solutions.

Proposition (Vieta for the monic quadratic). Let $x_{1}, x_{2}$ be the two roots (counted with multiplicity) of $x^{2} + p x + q = 0$ over a field $k$ . Then $x_{1} + x_{2} = - p$ and $x_{1} x_{2} = q$ .

Proof. The monic quadratic factors as $x^{2} + p x + q = (x - x_{1}) (x - x_{2})$ over an extension splitting the polynomial. Expanding the right-hand side: $(x - x_{1}) (x - x_{2}) = x^{2} - (x_{1} + x_{2}) x + x_{1} x_{2}$ . Comparing coefficients of $x^{1}$ and $x^{0}$ gives $- (x_{1} + x_{2}) = p$ and $x_{1} x_{2} = q$ .

Proposition (discriminant of a polynomial via Vieta). For a monic polynomial $f (x) = \prod_{i = 1}^{n} (x - r_{i})$ over a field, the discriminant $Δ (f) = \prod_{i < j} (r_{i} - r_{j})^{2}$ is symmetric in the roots, hence a polynomial expression in the coefficients of $f$ .

Proof sketch. Each permutation $σ \in S_{n}$ acts on the roots by $r_{i} \mapsto r_{σ (i)}$ . The expression $\prod_{i < j} (r_{i} - r_{j})^{2}$ is invariant under $σ$ because squaring removes the sign ambiguity in $r_{i} - r_{j}$ versus $r_{j} - r_{i}$ . By the fundamental theorem of symmetric functions, every symmetric polynomial in $r_{1}, \dots, r_{n}$ is a polynomial in the elementary symmetric functions $e_{1}, \dots, e_{n}$ , which by Vieta's formulas equal $\pm$ the coefficients of $f$ in alternating sign. Hence $Δ (f)$ is a polynomial in the coefficients. For $n = 2$ explicit computation gives $(r_{1} - r_{2})^{2} = (r_{1} + r_{2})^{2} - 4 r_{1} r_{2} = e_{1}^{2} - 4 e_{2} = p^{2} - 4 q = b^{2} - 4 a c$ after restoring the leading coefficient.

Proposition (Sylvester's law of inertia, statement). Let $Q : R^{n} \to R$ be a quadratic form. There exist non-negative integers $p, q$ with $p + q \leq n$ and a basis of $R^{n}$ in which $Q (v) = \sum_{i = 1}^{p} y_{i}^{2} - \sum_{j = 1}^{q} y_{p + j}^{2}$ where $v = \sum y_{i} e_{i}$ . The pair $(p, q)$ is independent of the choice of basis: any two diagonal forms of $Q$ have the same numbers of positive and negative coefficients.

Proof sketch. Existence of the diagonalisation: pick an orthonormal basis of eigenvectors of the matrix of $Q$ (spectral theorem), then rescale to absorb the absolute values of the nonzero eigenvalues. Independence: suppose $Q$ diagonalises as $\sum_{i = 1}^{p} y_{i}^{2} - \sum_{j = 1}^{q} y_{p + j}^{2}$ in one basis and $\sum_{i = 1}^{p^{'}} z_{i}^{2} - \sum_{j = 1}^{q^{'}} z_{p^{'} + j}^{2}$ in another. The subspace where the first diagonal form is non-negative has dimension $\geq p$ (the span of the first $p$ basis vectors), and the subspace where the second is strictly negative has dimension $\geq q^{'}$ (the span of the next-to-last $q^{'}$ basis vectors). By dimension counting, if $p > p^{'}$ these two subspaces share a nonzero vector on which $Q$ is both $\geq 0$ and $< 0$ , a contradiction. Hence $p \leq p^{'}$ , and by symmetry $p^{'} \leq p$ , so $p = p^{'}$ and likewise $q = q^{'}$ . (Stated without all details; see Lang Algebra Ch. XV.)

Proposition (binary conic discriminant). Let $C (x, y) = a x^{2} + b x y + c y^{2} + d x + ey + f$ be a degree-two polynomial in two variables over $R$ , with $(a, b, c) \neq = 0$ . The affine type of the conic ${C = 0}$ is determined by the sign of the binary discriminant $Δ_{C} = b^{2} - 4 a c$ : ellipse if $Δ_{C} < 0$ , parabola if $Δ_{C} = 0$ , and hyperbola if $Δ_{C} > 0$ .

Proof sketch. The principal $2 \times 2$ minor of the symmetric matrix $M_{2} = (a b /2 b /2 c)$ has determinant $det M_{2} = a c - b^{2} /4 = - Δ_{C} /4$ . By Sylvester's law of inertia, the binary quadratic form $a x^{2} + b x y + c y^{2}$ has signature $(2, 0)$ or $(0, 2)$ (definite) iff $det M_{2} > 0$ iff $Δ_{C} < 0$ , signature $(1, 0)$ or $(0, 1)$ (semi-definite, rank $1$ ) iff $det M_{2} = 0$ iff $Δ_{C} = 0$ , and signature $(1, 1)$ (indefinite) iff $det M_{2} < 0$ iff $Δ_{C} > 0$ . The three cases correspond to the affine types of conics with non-degenerate full $3 \times 3$ matrix: ellipse, parabola, and hyperbola, respectively, by reducing to canonical form via an affine change of coordinates that diagonalises the quadratic part and absorbs the linear part. (Stated for non-degenerate conics; degenerate cases — point conic, line pair, double line — are detected by the full $3 \times 3$ determinant of the homogeneous lifted form. See Audin Geometry Ch. VI.)

Connections [Master]

The discriminant $Δ = b^{2} - 4 a c$ proved here as the obstruction to having two distinct real roots is the simplest non-degenerate instance of the discriminant of a polynomial introduced in 00.01.03 and generalised in 01.01.07 (the determinant) and the algebra-strand discriminant of an algebraic number field. The conic discriminant in the Advanced results section is the binary-quadratic-form version, reappearing in the conic-section classification of 00.10.01 pending and the projective-conic classification of the algebraic-geometry strand. The discriminant detects repeated roots in every later algebra unit where the question arises, from the resolvent cubic of the quartic formula to the ramification of primes in algebraic number theory.
Completing the square — the substitution $y = x + b / (2 a)$ that removes the linear term — is the prototype of the depression substitutions used to derive the cubic and quartic formulas, and the prototype of the diagonalisation of quadratic forms by orthogonal change of basis introduced in 01.01.15 (bilinear and quadratic forms). Sylvester's law of inertia gives a complete invariant for real quadratic forms in $n$ variables, the signature $(p, q)$ , of which the $n = 1$ case is the sign of the leading coefficient and the $n = 2$ case is the conic discriminant. The substitution-and-reduce pattern recurs throughout the algebra and analysis strands: in the Lagrange resolvents that drive Galois theory, in the normal-form reductions of linear endomorphisms (Jordan form, rational canonical form), and in the eigenvalue diagonalisation used throughout the linear-algebra strand.
Vieta's formulas $x_{1} + x_{2} = - b / a$ and $x_{1} x_{2} = c / a$ are the prototype of the elementary-symmetric-function relations between the roots and coefficients of a polynomial, reappearing in 00.01.03 (polynomials) as the general identity expressing the coefficients as elementary symmetric functions of the roots, in Newton's identities and the theory of symmetric polynomials, and in the Galois-theoretic correspondence that organises [03.05.*] (Galois theory and field extensions). The symmetric-function framework is the algebraic shadow of the permutation-of-roots picture that underlies Galois's solubility criterion: a polynomial is solvable by radicals iff its Galois group is soluble.
The reduction of the quadratic equation over $C$ to the simple fact that every complex number has a square root is the simplest instance of the fundamental theorem of algebra (Gauss 1799), the statement that $C$ is algebraically closed. The fundamental theorem reappears in 00.01.03 (polynomials), is proved in [06.02.*] (complex analysis) using Liouville's theorem or the argument principle, and is the foundational result of complex algebraic geometry. The story that adjoining square roots to $R$ produces algebraic closure of $R$ is exceptional to the archimedean real field; over the $p$ -adic numbers $Q_{p}$ no such low-degree closure exists.

Historical & philosophical context [Master]

The systematic solution of quadratic equations is among the oldest results in mathematics. Babylonian scribes, working in the period circa 2000-1600 BCE, solved specific quadratics of the form $x^{2} + b x = c$ and $x^{2} = b x + c$ using what amounts to completing the square in geometric form. Problem texts on cuneiform tablets (notably BM 13901 and YBC 6967) record procedures for finding two numbers given their sum and product — the special case of Vieta's identities — and for solving rectangle-area problems that reduce to single-variable quadratics ^{[Lang — Basic Mathematics Ch. 3–4]}. The Babylonian method does not state a general formula in symbols (algebraic notation predates Viète by three and a half millennia) but operates on numerical instances by a fixed recipe that corresponds exactly to the modern derivation.

Muhammad ibn Mūsā al-Khwārizmī (~780–850 CE), working at the House of Wisdom in Baghdad under the patronage of the Abbasid caliph al-Ma'mūn, wrote the Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala (The Compendious Book on Calculation by Completion and Balancing) around 825 CE ^{[al-Khwārizmī — Kitāb al-jabr wa-l-muqābala]}. The Arabic word al-jabr — meaning completion or restoration, the operation of adding the same quantity to both sides of an equation — is the origin of the English word algebra. Al-Khwārizmī classified quadratic equations into six canonical forms (working only with non-negative coefficients and avoiding subtraction) and gave a complete geometric demonstration of the completing-the-square procedure for each form, including a literal square-region diagram that motivates the name. His treatise was translated into Latin in the twelfth century by Robert of Chester and Gerard of Cremona, transmitting algebra to medieval Europe.

The first full algebraic treatment in the modern sense appeared in Gerolamo Cardano's 1545 Artis magnae sive de regulis algebraicis liber unus (The Great Art, or the Rules of Algebra), published in Nuremberg ^{[Cardano — Ars Magna]}. Cardano's Ars Magna gave systematic procedures for the cubic (drawing on the prior unpublished work of Scipione del Ferro and Niccolò Tartaglia) and the quartic (due to Cardano's student Lodovico Ferrari), placing the quadratic case as the simplest instance of a general theory. Cardano was also the first European to take negative roots seriously as legitimate solutions and to consider expressions involving square roots of negative numbers (in his analysis of the casus irreducibilis of the cubic) — a step toward the eventual recognition of complex numbers.

François Viète (1540–1603), in his 1591 In artem analyticen isagoge (Introduction to the Analytic Art), introduced the use of letters for both knowns and unknowns in algebraic equations, the symbolic-algebra innovation that enables the modern statement of the quadratic formula in coefficients ^{[Viète — In artem analyticen isagoge]}. Viète also stated the relations between the roots and coefficients of a polynomial that now bear his name. The systematic theory of the discriminant and the resolvent constructions for higher-degree polynomials developed through the eighteenth century in the work of Lagrange (Réflexions sur la résolution algébrique des équations, 1770–71), Vandermonde, and Ruffini, culminating in Niels Henrik Abel's 1824 proof of the impossibility of a general radical formula for the quintic and Évariste Galois's 1832 manuscript identifying the group-theoretic obstruction. Carl Friedrich Gauss, in his 1801 Disquisitiones Arithmeticae, proved the quadratic reciprocity law that he called the aureum theorema — establishing the modular arithmetic of quadratic residues as the foundational case of class-field-theoretic reciprocity ^{[Gauss — Disquisitiones Arithmeticae]}. Gauss's 1799 doctoral dissertation had two years earlier supplied the first essentially complete proof that $C$ is algebraically closed, completing the story that began with the quadratic case.

Bibliography [Master]

[object Promise]

Prerequisites

00.01.03

Tier anchors

beginner: Lang Basic Mathematics Ch. 3; Strogatz-style parabola-crossing-the-axis picture
intermediate: Lang Basic Mathematics Ch. 3–4; Apostol Calculus Vol. 1 §I.4; Artin Algebra Ch. 11
master: al-Khwārizmī ~825 Al-jabr; Cardano 1545 Ars Magna; Viète 1591 In artem analyticen isagoge; Gauss 1801 Disquisitiones Arithmeticae; Lang Algebra Ch. IV §1

References

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Lang — Basic Mathematics · Ch. 3–4, quadratic equations and completing the square
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Apostol — Calculus Vol. 1 · §I.4, quadratic equations and inequalities
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Artin — Algebra · Ch. 11, polynomial rings and quadratic resolvents
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al-Khwārizmī — Kitāb al-jabr wa-l-muqābala · ~825 CE, systematic completion-of-the-square method
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Cardano — Ars Magna · 1545, Chapters I–XI, cubic and quartic formulas
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Viète — In artem analyticen isagoge · 1591, literal coefficient notation and Vieta's formulas
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Gauss — Disquisitiones Arithmeticae · 1801, §IV, quadratic reciprocity
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Lang — Algebra · Ch. IV §1, polynomial roots over a field

Lean module

Codex.Precalc.Equations.QuadraticFormula

Mathlib gap

Mathlib supplies the discriminant via Mathlib.Algebra.QuadraticDiscriminant where discrim a b c is b^2 - 4ac, the characterisation quadratic_eq_zero_iff giving the root pair in terms of a square root of the discriminant over a field of characteristic not two, the trichotomy discrim_lt_zero / discrim_eq_zero / discrim_pos for real-coefficient quadratics, and the Vieta-style identities sum_roots and prod_roots packaged in the same module. What is not packaged as a single named lemma is the textbook completing-the-square identity a*x^2 + b*x + c = a*(x + b/(2a))^2 - (b^2 - 4ac)/(4a) as a pointwise polynomial identity, which the companion module states alongside the formula and the discriminant trichotomy. The human reviewer signs off on the coverage claim.

Reviewer

TBD

Estimated time

beginner: 14m
intermediate: 30m
master: 50m