Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 with the quadratic formula, including the discriminant, the exact radical form of the roots, and a substitution check.
Calculator
- Discriminant analysis
- Two real / repeated / complex roots
- Exact radical form
- Substitution check
Result & calculation receipt
- Method
- Quadratic formula with discriminant analysis (two distinct real roots)
- Formula
x = (−b ± √(b² − 4ac)) / (2a)- Substitution
- x = (−-5 ± √1) / 2
- Steps
- Discriminant Δ = b² − 4ac = -5² − 4·1·6 = 1.
- Δ > 0 → two distinct real roots.
- x = (−b ± √Δ) / (2a) = (5 ± √1) / 2.
- Verification
- Check x=3: 0 ≈ 0; x=2: 0 ≈ 0.
- Exact result
- x = 3 or x = 2
- Decimal result
- root1: 3 · root2: 2
- Engine
- math-solvers v1.4.0
How this calculator works
The calculator first computes the discriminant b² − 4ac, which decides everything: a positive value gives two distinct real roots, zero gives one repeated real root, and a negative value gives a pair of complex-conjugate roots. It then applies x = (−b ± √Δ)/(2a), keeping the answer in exact radical form when the discriminant is not a perfect square and giving decimals as well.
Worked example
For x² − 5x + 6 = 0 the discriminant is 25 − 24 = 1, so there are two real roots: x = (5 ± 1)/2 = 3 or 2. For x² − 4x + 4 = 0 the discriminant is 0, giving the repeated root x = 2. For x² + 1 = 0 the discriminant is −4, so the roots are the complex conjugates ±i. Each root is substituted back to confirm it makes the equation zero.
Key insight
The discriminant is the quadratic’s fingerprint: its sign tells you the number and type of roots before you finish solving. That is why "b² − 4ac" is worth memorizing — it turns a solving problem into a quick classification.
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) / (2a). The ± produces the two roots.
What does the discriminant tell me?
The discriminant b² − 4ac determines the roots: positive means two distinct real roots, zero means one repeated real root, and negative means two complex-conjugate roots. The calculator classifies it for you.
What are complex roots?
When the discriminant is negative, the square root involves √(−1) = i, so the roots have the form p ± qi. They come in conjugate pairs and are real numbers only in the sense of the complex plane.
What is exact radical form?
When the discriminant is not a perfect square, the roots contain a square root, like (1 ± √5)/2. The calculator keeps that exact form instead of only giving a rounded decimal.
Why must a be non-zero?
If a = 0 the x² term vanishes and the equation is linear, not quadratic. The calculator directs you to the linear equation solver in that case.
Methodology & verification
The discriminant Δ = b² − 4ac is computed and used to classify the roots: Δ > 0 gives two distinct real roots, Δ = 0 a repeated real root, Δ < 0 a complex-conjugate pair. Roots come from x = (−b ± √Δ)/(2a), kept in exact radical form when Δ is not a perfect square and as decimals otherwise. a = 0 is rejected as non-quadratic. Real roots are verified by substitution; complex roots are verified against the sum (−b/a) and product (c/a) of roots.
Engine: math-solvers v1.4.0 · Last reviewed: · Every result is computed client-side and never sent to a server. See the verification methodology and the scientific calculator validation.