Quadratic Formula Calculator
Solve quadratic equations ax² + bx + c = 0 and find real or complex roots with step-by-step solutions.
Quadratic Equation: ax² + bx + c = 0
Example
For the equation x² - 5x + 6 = 0:
- a = 1, b = -5, c = 6
- Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1
- x = [5 ± √1] / 2 = [5 ± 1] / 2
- Solutions: x₁ = 3, x₂ = 2
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How to Use This Calculator
- Write your quadratic equation in standard form: ax² + bx + c = 0
- Enter the coefficient 'a' (the number multiplying x²)
- Enter the coefficient 'b' (the number multiplying x)
- Enter the constant term 'c' (the number without any variable)
- Click Calculate to find the solutions and additional properties like vertex and discriminant
Formula
For ax² + bx + c = 0: x = [-b ± √(b² - 4ac)] / (2a). Discriminant Δ = b² - 4ac determines solution type: Δ > 0 (two real roots), Δ = 0 (one repeated root), Δ < 0 (two complex roots). Vertex: (-b/2a, f(-b/2a))
Frequently Asked Questions
What is the discriminant?▼
The discriminant is b² - 4ac. It determines the nature of the roots: positive means two real solutions, zero means one repeated solution, and negative means two complex (imaginary) solutions.
What does it mean when a parabola opens upward or downward?▼
When coefficient 'a' is positive, the parabola opens upward (U-shape). When 'a' is negative, it opens downward (∩-shape). This affects whether the vertex is a minimum or maximum point.
What is the vertex of a parabola?▼
The vertex is the turning point of the parabola - either the minimum point (if opening upward) or maximum point (if opening downward). It's located at x = -b/(2a).
Can the quadratic formula be used for all quadratic equations?▼
Yes! The quadratic formula works for all quadratic equations where a ≠ 0. It always produces the correct solutions, whether they're real, repeated, or complex numbers.