Quadratic Formula Calculator
Solve quadratic equations instantly
Reference table
| Coefficient a (x²) | Root x₁ | Equation | Discriminant (Δ) |
|---|---|---|---|
| 0.5 | 5.2361 | 0.5x² - 3x + 2 = 0 | 5 |
| 1 | 2 | x² - 3x + 2 = 0 | 1 |
| 1.5 | 1 + 0.5774i | 1.5x² - 3x + 2 = 0 | -3 |
| 2 | 0.75 + 0.6614i | 2x² - 3x + 2 = 0 | -7 |
| 2.5 | 0.6 + 0.6633i | 2.5x² - 3x + 2 = 0 | -11 |
| 3 | 0.5 + 0.6455i | 3x² - 3x + 2 = 0 | -15 |
| 4 | 0.375 + 0.5995i | 4x² - 3x + 2 = 0 | -23 |
| 5 | 0.3 + 0.5568i | 5x² - 3x + 2 = 0 | -31 |
| 6 | 0.25 + 0.5204i | 6x² - 3x + 2 = 0 | -39 |
| 7 | 0.2143 + 0.4897i | 7x² - 3x + 2 = 0 | -47 |
| 8 | 0.1875 + 0.4635i | 8x² - 3x + 2 = 0 | -55 |
| 9 | 0.1667 + 0.441i | 9x² - 3x + 2 = 0 | -63 |
| 10 | 0.15 + 0.4213i | 10x² - 3x + 2 = 0 | -71 |
| 12 | 0.125 + 0.3886i | 12x² - 3x + 2 = 0 | -87 |
| 15 | 0.1 + 0.3512i | 15x² - 3x + 2 = 0 | -111 |
📊 Discriminant Cases
- Δ > 0Two distinct real roots
- Δ = 0One repeated real root (x = -b/2a)
- Δ < 0Two complex conjugate roots
- Δ = perfect squareRational roots (integer a, b, c)
Quick Tips
- Δ > 0: Two distinct real roots
- Δ = 0: One repeated real root
- Δ < 0: Two complex conjugate roots
- Vertex x = -b/(2a)
What is the Quadratic Formula?
Understanding the Discriminant
Important Properties
- The coefficient 'a' cannot be zero (otherwise it's linear, not quadratic)
- Vieta's formulas relate roots to coefficients: sum = -b/a, product = c/a
- The vertex (h,k) represents the minimum (a>0) or maximum (a<0) point
- Vertex form y = a(x-h)² + k makes graphing easier
- Complex roots always come in conjugate pairs
Real-World Applications
- Projectile motion: height follows h = h₀ + v₀t − ½gt², a quadratic in time
- Business: revenue and profit curves are often parabolas — the vertex gives the optimal price or output
- Geometry: area, fencing, and border problems reduce to quadratic equations
- Engineering: parabolic arches, suspension-bridge cables, and satellite dishes are designed with quadratics
- Road safety: braking distance grows with the square of speed — double the speed means roughly four times the distance
- Computer graphics: Bézier curves and ray–sphere intersections solve quadratic equations every frame
Example Calculations
Step-by-step solutions for different cases
Two Real Roots
- Equation: x² - 5x + 6 = 0
- a=1, b=-5, c=6
- Δ = 25 - 24 = 1 > 0
- x = (5 ± 1) / 2
x₁ = 3, x₂ = 2
Complex Roots
- Equation: x² + 2x + 5 = 0
- a=1, b=2, c=5
- Δ = 4 - 20 = -16 < 0
- x = (-2 ± 4i) / 2
x₁ = -1+2i, x₂ = -1-2i
Frequently Asked Questions
What does the discriminant tell us?
The discriminant (b² - 4ac) reveals the nature of roots: positive means two real roots, zero means one repeated root, negative means two complex roots.
Can I solve any quadratic with this formula?
Yes! The quadratic formula works for any quadratic equation ax² + bx + c = 0 where a ≠ 0, regardless of whether the roots are real or complex.
What is the vertex of a parabola?
The vertex is the turning point of the parabola. For y = ax² + bx + c, the vertex is at x = -b/(2a). It's the minimum point if a > 0, maximum if a < 0.
What are Vieta's formulas?
Vieta's formulas relate roots to coefficients: the sum of roots equals -b/a, and the product of roots equals c/a. These work even for complex roots.
How do I convert to vertex form?
Complete the square: y = a(x² + (b/a)x) + c becomes y = a(x + b/2a)² + (c - b²/4a). The vertex is (-b/2a, c - b²/4a).
What if 'a' equals zero?
If a = 0, it's no longer quadratic—it becomes a linear equation bx + c = 0, which has one solution: x = -c/b (if b ≠ 0).