Quadratic equation calculator

Definition

A quadratic equation over the reals is an equation of the form ax² + bx + c = 0, where a, b, c ∈ ℝ and a ≠ 0. The unknowns are the values of x that make the degree-two polynomial vanish. The discriminant Δ = b² − 4ac decides how many real roots exist. The quadratic formula x = (−b ± √Δ) / (2a) gives them explicitly when Δ ≥ 0. This page implements exactly that path in the browser’s floating-point arithmetic and rounds displayed roots to six decimal places (round(x · 10⁶) / 10⁶).

When a = 0 the equation degenerates to bx + c = 0: if b ≠ 0 then x = −c/b; if b = c = 0 — infinitely many solutions; if b = 0 and c ≠ 0 — none.

Historical and conceptual context

Quadratic procedures were known in Babylonia. Al-Khwarizmi (9th century) systematized completing the square. In Europe the discriminant form was popularized by Cardano (16th century). Geometric square treatments also appear in Euclid’s Elements (Book II).

Conceptually a quadratic equation joins algebra (polynomial roots) with geometry (intersections of the parabola y = ax² + bx + c with the x-axis). The vertex sits at x = −b/(2a) — the same place as the double root when Δ = 0. In physics, projectile motion in vacuum is parabolic; optics and mechanics reuse the same quadratic structures. This calculator does not solve higher-degree equations or systems — only ax² + bx + c = 0 (or the linear degeneration).

Formal properties

Coefficient domain: a, b, c ∈ ℝ. For a ≠ 0: Δ = b² − 4ac. If Δ > 0, two distinct real roots; if Δ = 0, one double root; if Δ < 0, no roots in ℝ (roots exist in ℂ, outside this page). Vieta’s formulas: x₁ + x₂ = −b/a, x₁ · x₂ = c/a (for two roots, including a double root read as x₁ = x₂).

The parabola opens upward when a > 0 and downward when a < 0. The axis of symmetry is x = −b/(2a).

Full derivation

Step 1. Assume a ≠ 0. Divide by a: x² + (b/a)x + c/a = 0.

Step 2. Complete the square: x² + (b/a)x = (x + b/(2a))² − (b/(2a))². Hence (x + b/(2a))² = (b² − 4ac)/(4a²) = Δ/(4a²).

Step 3. When Δ ≥ 0: x + b/(2a) = ± √Δ / (2a), so x = (−b ± √Δ) / (2a). That is the same formula the calculator computes.

Step 4 (Vieta). For two roots, sum and product follow from expanding a(x − x₁)(x − x₂) = ax² − a(x₁+x₂)x + a x₁ x₂ and matching ax² + bx + c.

Step 5 (degeneration). When a = 0 the equation is no longer degree 2; the linear path bx + c = 0 is handled separately, with no Δ and no parabola chart.

Special cases

Examples: 1, −3, 2 → Δ = 1, x = 1 and 2; 1, 2, 1 → x = −1; 1, 0, 1 → no real roots; 2, 0, −8 → x = −2 and 2; 0, 2, −4 → x = 2.

When b = 0 and Δ > 0, roots are symmetric about zero. When c = 0 and a ≠ 0, one root is x = 0.

Numerical notes

The calculator computes delta = b·b − 4·a·c, then roots with √(delta) and division by 2·a. The comparison delta > 0 / === 0 / < 0 is raw in floating-point arithmetic — when Δ is extremely close to zero, rounding can matter. Display uses rounding to 10⁶. A period and a comma are equivalent in the parser.

The SVG chart ranges around the roots (or [−3, 3] with no zeros).

The result panel shows a, b, c, Δ and the roots x₁ and x₂ when they exist. When Δ > 0 both roots differ; when Δ = 0 the calculator shows the double root; when Δ < 0 the message reports no real roots — without switching to complex numbers. The parabola chart y = ax² + bx + c appears only when a ≠ 0 and helps you see the x-axis intersections.

Vieta’s formulas give a quick check: the sum of roots should be near −b/a and the product near c/a. For 1, −3, 2 the sum is 3 = −(−3)/1 and the product is 2 = 2/1. For 1, 2, 1 the double root −1 satisfies (−1)+(−1) = −2 and (−1)·(−1) = 1. That check is not a separate form mode — it follows from the same quadratic formula.

When a = 0 the page does not draw a parabola and does not compute Δ. The linear branch bx + c = 0 is intentional: empty fields may be treated as zero, but the linear result must still be finite. A higher-degree equation, a two-equation system, or a quadratic inequality lies outside this calculator.

Related notions and sources

Related: linear equation, linear function value, difference of squares, square root. Sources: Wikipedia — Quadratic equation; MathWorld — Quadratic Equation; al-Khwarizmi; Euclid, Elements II.

In school practice, compute Δ first, then the roots, and finally check by substituting into ax² + bx + c. For 1, −3, 2: 1 − 3 + 2 = 0 and 4 − 6 + 2 = 0. Educational tool; it does not replace a textbook or a teacher’s check.