Solve exact rational quadratic coefficients, distinguish real and complex roots and inspect an interactive polynomial graph.
01 / PURPOSE-SPECIFIC INPUTS
Decimals, bounded scientific notation and exact fractions are accepted. Linear, constant and identity cases are classified explicitly.
02 / RESULT & INTERACTIVE INSPECTION
real. Real x-intercepts: 2, 3. Vertex (2.5, -0.25). Zoom 1×.
| Step / method | Exact working and labeled approximations |
|---|---|
| Original equation | 1x² + (-5)x + (6) = 0 |
| Exact coefficients | a = 1, b = -5, c = 6 |
| Quadratic formula | x = [−b ± √(b² − 4ac)] / (2a) |
| Substitute coefficients | x = [−(-5) ± √((-5)² − 4 × (1) × (6))] / [2 × (1)]. |
| Exact discriminant | b² = 25; 4ac = 24; Δ = 25 − (24) = 1. Exact rational sign gives real roots. |
| Simplify the square root | √(1) = 1. |
| Both exact roots | x₁ = 2; x₂ = 3. |
| Decimal approximations | x₁ ≈ 2; x₂ ≈ 3. Rounded values are not exact roots. |
| Verify root 1 exactly | Substitute x = 2: (1)(2)² + (-5)(2) + (6). Exact rational evaluation = 0. |
| Verify root 2 exactly | Substitute x = 3: (1)(3)² + (-5)(3) + (6). Exact rational evaluation = 0. |
| Factoring | 1 × (x − (2))(x − (3)) |
| Completing the square · 1 | Divide by a = 1 ≠ 0: x² + (-5)x + (6) = 0. |
| Completing the square · 2 | Move the constant: x² + (-5)x = -6. |
| Completing the square · 3 | Add (b/(2a))² = (-5/2)² = 25/4 to both sides. |
| Completing the square · 4 | (x + (-5/2))² = 25/4 − (6) = 1/4 = Δ/(4a²). |
| Completing the square · 5 | Take exact real square roots: x + (-5/2) = ± 1/2; x = 5/2 ± 1/2. |
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Enter exact decimal, scientific or fraction coefficients. Solve using exact discriminant classification. Inspect roots, vertex and the graph’s finite display resolution. Read the worked explanation and scope alongside every result.
Solve exact rational quadratic coefficients, distinguish real and complex roots and inspect an interactive polynomial graph. Interactive controls inspect the calculated model without silently changing submitted assumptions.
Quadratic Equation Solver inputs and calculations remain in browser memory. No external processing API receives entered dates, measurements, coefficients or financial values. Ordinary website page requests may still be logged. Clipboard access occurs only when you choose Copy Result.
Bounded decimal, scientific-notation and rational fraction inputs are parsed exactly before classification. Nonzero numeric coefficient magnitudes are bounded from 1e−100 to 1e100.
The discriminant b²−4ac is computed with exact rational arithmetic. Positive, zero and negative signs classify distinct real, repeated and complex-conjugate roots without a floating near-zero guess.
Perfect-square rational discriminants produce exact rational roots. Other roots receive numerical approximations, with stable real-root evaluation to reduce cancellation. Displayed precision does not turn an approximation into a symbolic exact value.
The equation is solved as linear if b is nonzero. If both a and b are zero, c = 0 is an identity with all real values; nonzero c is an inconsistent constant equation.
It plots the real polynomial but shows no real x-intercepts for complex roots. Vertex and symmetry are separate from root existence; finite graph sampling does not replace exact discriminant classification.
Yes. Switch between vertex/root-centered and origin-centered views, zoom, pan or reset the graph. Extremely large or closely spaced values can exceed floating display resolution, while the exact text result remains authoritative.