Quadratic Formula Calculator — Solve ax²+bx+c=0 Instantly
Solve any quadratic equation ax²+bx+c=0 with the quadratic formula. Get both roots (x₁, x₂) and the discriminant (Δ = b²−4ac) instantly — handles real, repeated, and complex roots.
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When to use this calculator
- Determine the two moments when a ball thrown upward at 20 m/s from a 5 m platform hits the ground: −5t² + 20t + 5 = 0.
- Find break-even quantities for a business with revenue R(x) = −2x² + 80x − 600 by solving −2x² + 80x − 600 = 0.
- Calculate dimensions of a rectangular garden with area 36 ft² and perimeter 26 ft, reducing to x² − 13x + 36 = 0.
- Verify whether a conic section has real intersections by checking the discriminant sign before plotting.
- Solve for the resonance frequency in an RLC circuit equation that simplifies to a quadratic.
Famous & Real-World Quadratic Equations (worked)
Recognizable second-degree equations from math, physics, finance and geometry — with discriminant and exact roots.
| Equation | a | b | c | Δ = b²−4ac | Roots (x₁, x₂) | Where it appears |
|---|---|---|---|---|---|---|
| x² − x − 1 = 0 | 1 | −1 | −1 | 5 | 1.618, −0.618 | Golden ratio φ = (1+√5)/2 |
| x² − 13x + 36 = 0 | 1 | −13 | 36 | 25 | 9, 4 | Rectangle: area 36, perimeter 26 |
| −2x² + 80x − 600 = 0 | −2 | 80 | −600 | 1,600 | 10, 30 | Business break-even quantities |
| x² − 4x + 1 = 0 | 1 | −4 | 1 | 12 | 3.732, 0.268 | 2 ± √3 (irrational roots) |
| x² − 6x + 9 = 0 | 1 | −6 | 9 | 0 | 3 (double root) | Perfect-square trinomial (x−3)² |
| x² + x + 1 = 0 | 1 | 1 | 1 | −3 | −½ ± i·√3/2 | Complex conjugate roots (Δ<0) |
| −4.9t² + 15t + 20 = 0 | −4.9 | 15 | 20 | 617 | 4.07 s (t>0) | Projectile from 20 m at 15 m/s |
Δ>0 → two real roots; Δ=0 → one repeated root; Δ<0 → two complex conjugate roots. Verify any pair with Vieta: x₁+x₂=−b/a and x₁·x₂=c/a. Projectile root keeps only the positive (physical) time.
How it works
How the Quadratic Formula Works
The quadratic formula is derived by completing the square on ax² + bx + c = 0:
Step 1 — Compute the discriminant:
Δ = b² − 4ac
Step 2 — Apply the quadratic formula:
x₁ = (−b + √Δ) / (2a)
x₂ = (−b − √Δ) / (2a)
Step 3 — Interpret Δ:
Δ > 0 → two distinct real roots
Δ = 0 → one repeated real root: x = −b / (2a)
Δ < 0 → two complex conjugate roots: x = −b/(2a) ± i·√|Δ|/(2a)Vieta's formulas let you double-check your answer:
x₁ + x₂ = −b / a
x₁ · x₂ = c / a---
Discriminant Reference Table
| Discriminant (Δ) | Root type | Example equation | Roots |
|---|---|---|---|
| Δ > 0 (perfect square) | 2 rational real roots | x² − 5x + 6 = 0 (Δ=1) | x₁=3, x₂=2 |
| Δ > 0 (non-perfect square) | 2 irrational real roots | x² − 4x + 1 = 0 (Δ=12) | x=2±√3 |
| Δ = 0 | 1 repeated real root | x² − 6x + 9 = 0 (Δ=0) | x=3 (double) |
| Δ < 0 | 2 complex conjugate roots | x² + x + 1 = 0 (Δ=−3) | x=−½ ± i√3/2 |
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Quick-Reference: Common Equations Solved
| Equation | a | b | c | Δ | x₁ | x₂ |
|---|---|---|---|---|---|---|
| x²−5x+6=0 | 1 | −5 | 6 | 1 | 3 | 2 |
| x²−6x+9=0 | 1 | −6 | 9 | 0 | 3 | 3 (double) |
| x²+x+1=0 | 1 | 1 | 1 | −3 | −½+i√3/2 | −½−i√3/2 |
| 2x²−3x−2=0 | 2 | −3 | −2 | 25 | 2 | −0.5 |
| x²−4x+1=0 | 1 | −4 | 1 | 12 | 2+√3≈3.732 | 2−√3≈0.268 |
| −x²+4x−3=0 | −1 | 4 | −3 | 4 | 3 | 1 |
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Worked Examples
Example 1 — Two distinct rational roots
Equation: x² − 5x + 6 = 0 (a=1, b=−5, c=6)
Example 2 — One repeated root (double root)
Equation: x² − 6x + 9 = 0 (a=1, b=−6, c=9)
Example 3 — Complex roots
Equation: x² + 2x + 5 = 0 (a=1, b=2, c=5)
Example 4 — Projectile motion
A ball is thrown upward at 20 m/s from a 5 m height: −5t² + 20t + 5 = 0
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Common Mistakes to Avoid
1. a ≠ 0 is required. If a=0 the equation is linear, not quadratic; this calculator will warn you.
2. Sign of b in the numerator. The formula uses −b. If b=−5, then −b=+5. A sign error shifts both roots by 2b/a.
3. 4ac includes all signs. With a=2, c=−2: 4ac = 4·2·(−2) = −16, so Δ=9−(−16)=25, not 9−16=−7.
4. Divide the full numerator by 2a. The correct form is (−b ± √Δ) / (2a), not −b ± (√Δ / 2a).
5. Complex roots are still solutions. When Δ<0, the equation has no real roots but always has two complex conjugate solutions in ℂ.
6. Rearrange to standard form first. From 3x²=5x−2, rearrange to 3x²−5x+2=0 (c=+2), then identify a=3, b=−5, c=2.
Worked Example: x² − 5x + 6 = 0
Frequently asked questions
What does the discriminant tell me about the graph of a parabola?
How do I use the quadratic formula step by step?
Can I use this calculator when a is negative?
Why does completing the square give the same result as the quadratic formula?
What are Vieta's formulas and why are they useful?
How do I handle complex roots in practice?
Is there a formula for cubic or higher-degree equations?
What happens when b=0 or c=0?
How does the quadratic formula apply to physics problems?
Sources & references
Methodology & trust
Calculadora de matemática revisada por el equipo editorial de Hacé Cuentas, contrastada con NIST Digital Library of Mathematical Functions — §1.11 Algebraic and Analytic Methods, según nuestra política editorial y metodología.
Última revisión: June 22, 2026. Los parámetros se verifican periódicamente con las fuentes citadas.
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Indicative results. For critical decisions, consult a professional.
Rodríguez, M. (2026). Quadratic Formula Calculator — Solve ax²+bx+c=0 Instantly. Hacé Cuentas. https://hacecuentas.com/quadratic-equation-roots-discriminant
Contenido bajo licencia CC-BY 4.0 — reutilizable citando la fuente con enlace a Hacé Cuentas.