What does the discriminant tell me?

b² − 4ac determines the nature of the roots. Positive: two distinct real roots (parabola crosses x-axis twice). Zero: one double real root (parabola touches the axis). Negative: two complex conjugate roots (parabola doesn't cross the x-axis at all).

When do quadratics have complex roots?

When the discriminant is negative — the parabola doesn't intersect the x-axis. Real-world physical models that produce complex roots typically mean 'this configuration has no real solution' (e.g. a projectile that never reaches a given height).

How do I factor a quadratic from the roots?

If the roots are r₁ and r₂, the quadratic factors as a(x − r₁)(x − r₂). Useful for going from solutions back to a polynomial form for further manipulation or for matching to a known canonical form.

discriminant · vertex · complex · local

Solve a quadratic equation

Type a, b, c and read x₁, x₂, the discriminant, and the parabola vertex. Handles real, double, and complex roots — plus degenerate cases.

Quadratic Equation Solver

Verdict

two

case

discriminant

Vertex x

2.5

axis

Output

59 B

ready

input · ax² + bx + c = 0

1x² + -5x + 6 = 0

abc

output · roots

Δ = b² − 4ac = 1(two real roots)

x₁

x₂

vertex: (2.5, -0.25)

Quick Start

How the quadratic formula works

x = (−b ± √(b² − 4ac)) / 2a. The discriminant Δ tells you whether you get two real roots, one double root, or two complex conjugates.

01
Type a, b, c
Coefficients can be any finite number, positive or negative.
Inputs are treated as UTF-8 throughout the pipeline and validated client-side as you type — nothing dispatches until the value is well-formed.
inpututf-8client-sidevalidated
abc · text1,-5,6 · set
```
x² − 5x + 6 = 0
```
loaded · parsedclient-side
02
Read Δ
Discriminant tells the kind: positive → two real, zero → double, negative → complex.
Each toggle re-computes through a memoised hook, so flipping options never resets your input and the UI stays snappy on large inputs.
deltaoptionslivememoisedtoggles
Δ · memoised1 · live
```
Δ > 0
```
options set · memoisedlive
03
Inspect roots
Roots appear in big mono type, plus the parabola's vertex.
Randomness — where used — comes from crypto.getRandomValues, never Math.random. Heavy work sits in useMemo.
rootscomputeinstantlocaldeterministic
x · instant2, 3 · computed
```
x₁ = 2
x₂ = 3
```
computed · instantinstant
04
Copy result
One click pastes the full readable summary into your clipboard.
Copy writes through navigator.clipboard and downloads use a Blob — zero network traffic. Same input plus same options always yield the same bytes.
copydownloadzero-networkreproducible
ok · copiedtxt · copied
```
Copied
```
shipped · ready to shipsuccess

Pro tips

Pro tips for everyday utilities

Reach for crypto.getRandomValues

Cryptographically random for IDs, tokens, salts. Math.random is for animation and games, not security.

Sort IDs lexicographically

UUID v4 has no time ordering. Use ULID or UUID v7 if you need sorted IDs.

Mock data with seeds

Faker-style libraries can be seeded so tests are reproducible. Embed the seed in the fixture file.

Reference cards beat memory

Cheatsheets for Vim, Git, Docker, Linux signals are faster than man pages when you’re mid-flow.

Browser APIs vary

Touch events, gamepads, and clipboard differ across vendors. Feature-detect before use.

Persist in the right place

localStorage for user preferences, sessionStorage for transient state, IndexedDB for structured data.

Use Cases

When to solve a quadratic

Physics, engineering, finance, math homework.

Projectile motion

Compute when y(t) = 0 from kinematic equations. Random values come from crypto.getRandomValues — cryptographically secure for tokens, salts, and session IDs. Math.random is deterministic enough for animations but predictable enough to be a security bug.

CSPRNG-backed

Finance break-even

Find break-even quantity from a quadratic profit model. Pair with hashids, ULID, or UUID v7 when you need both random and sortable — useful for primary keys, public-facing IDs, and event log ordering without a sequence column.

reproducible

Geometry

Solve for x from area or distance equations. Output is deterministic given the same seed, so fixtures and tests stay reproducible across CI runs and contributor machines. Embed the seed in the fixture file for full replay.

snappy

Curve fitting

Find x-intercepts of a parabolic fit instantly. Heavy work runs through useMemo so the UI stays snappy on large inputs. Same input + same options yields identical bytes — the kind of stability snapshot tests rely on.

feature-detected

Engineering

Resonance and damping ratios reduce to a quadratic root. Storage tools persist to the right backend: localStorage for user preferences, sessionStorage for transient state, IndexedDB for structured data. Pick the right shelf and quota worries disappear.

deterministic

Math homework

Verify your by-hand solution and read the discriminant. Browser feature detection: clipboard, fullscreen, gamepad, and Web Bluetooth APIs gracefully fall back when unavailable. No surprise white screens on Safari or older Firefox.

production-ready

FAQ

Frequently asked questions

Why do I see scientific notation?

When values exceed 10¹² or fall below 10⁻⁴ we switch to scientific notation for readability.

What if a = 0?

We solve the linear equation bx + c = 0. If b is also zero we report the degenerate case.

Are complex roots exact?

They use double-precision floats — fine to many decimal places but not symbolic.

Where does vertex come from?

The parabola vertex is at x = −b/(2a); we plug into y for vy.

Is anything sent to a server?

No — solver runs locally in your browser.

Suggest a tool

Something missing?

Tell us what you need. We ship the top requests every two weeks.

47 in queue·8 shipped this month

Request queued. Thanks!