Input
InstantAssumes A·x + B·y + C = 0.
Assumes y = a(x − h)² + k; converts to y = ax² + bx + c.
Assumes y = a(x − r₁)(x − r₂); converts to y = ax² + bx + c.
Outputs (x − h)² + (y − k)² = r².
Assumes x² + y² + D·x + E·y + F = 0; converts to (x − h)² + (y − k)² = r².
Result
Fill inputs and click Convert to see the standard form and steps.
Formula used
- Line standard form: A·x + B·y + C = 0, with integer A, B, C, gcd(|A|,|B|,|C|)=1, and A ≥ 0.
- From slope–intercept: y = m x + b ⇒ m x − y + b = 0 ⇒ scale to integers ⇒ normalize gcd, sign.
- From point–slope: y − y₁ = m(x − x₁) ⇒ m x − y + (−m x₁ + y₁) = 0 ⇒ normalize.
- From two points: A = y₁ − y₂, B = x₂ − x₁, C = x₁y₂ − x₂y₁ ⇒ normalize.
- Quadratic: Vertex y = a(x−h)² + k ⇒ y = a x² − 2ah x + (ah² + k).
- Quadratic intercepts: y = a(x−r₁)(x−r₂) ⇒ y = a x² − a(r₁+r₂)x + a r₁ r₂.
- Circle general: x² + y² + D x + E y + F = 0 ⇒ (x + D/2)² + (y + E/2)² = (D²+E²)/4 − F.
How to use
- Choose Equation Type and the Input Form.
- Enter parameters. Decimals are fine; fractions are normalized.
- Click Convert to see standard form and derivation.
- Click Add row to append the result to the table.
- Use Download CSV or Download PDF to export the table.
Results table
| # | Timestamp | Type | Input form | Inputs | Standard form | Details |
|---|---|---|---|---|---|---|
| 1 | 2026-01-25 22:03:01 | Line | Two points | (1,2) and (4,8) | 2x − y = 0 | A=2, B=−1, C=0 |
| 2 | 2026-01-25 22:03:01 | Quadratic | Vertex | a=1, h=2, k=3 | y = x² − 4x + 7 | a=1, b=−4, c=7 |
| 3 | 2026-01-25 22:03:01 | Circle | General | D=−4, E=6, F=−12 | (x − 2)² + (y + 3)² = 25 | Center(2, −3), r=5 |
Example rows are prefilled for reference. Add your own using Add row.
FAQs
We use A·x + B·y + C = 0 with integer coefficients, gcd(|A|,|B|,|C|)=1, and A ≥ 0 for a consistent, unique representation.
Yes. Decimals are internally converted to rational numbers. We then scale to integer coefficients and normalize the greatest common divisor.
You can enter standard y = ax² + bx + c, vertex y = a(x−h)² + k, or intercept y = a(x−r₁)(x−r₂). We output y = ax² + bx + c.
From x² + y² + D x + E y + F = 0 we complete the square to (x − h)² + (y − k)² = r² and provide center and radius.
Use the buttons above the table to download CSV or generate a PDF snapshot of the table for reports or records.
Data: Common Conversions
Reference examples showcasing inputs and their normalized standard forms.
| # | Type | Input | Standard Form | Notes |
|---|---|---|---|---|
| 1 | Line | y = 2x − 3 | 2x − y − 3 = 0 | A=2, B=−1, C=−3; gcd=1 |
| 2 | Line | Through (−1,4), slope 3/2 | 3x − 2y + 11 = 0 | Scaled to integers, A ≥ 0 |
| 3 | Quadratic | y = (x − 2)² + 1 | y = x² − 4x + 5 | Expanded from vertex form |
| 4 | Quadratic | y = 2(x − 1)(x − 5) | y = 2x² − 12x + 10 | Intercept to standard |
| 5 | Circle | x² + y² − 6x + 8y − 11 = 0 | (x − 3)² + (y + 4)² = 36 | Center (3, −4), r = 6 |
Data: Normalization Summary
Typical coefficient normalization outcomes across representative inputs.
| Metric | Lines | Quadratics | Circles |
|---|---|---|---|
| Inputs needing integer scaling | 67% | 54% | 38% |
| Cases with gcd > 1 reduced | 49% | 41% | 22% |
| Outputs achieving A ≥ 0 after sign flip | 35% | — | — |
| Average terms simplified per expression | 2.1 | 2.7 | 1.6 |
Percentages are illustrative for documentation; actual runs depend on your data.