Quadratic Formula Calculator
Solve quadratic equations ax² + bx + c = 0 with discriminant analysis, real roots, and complex numbers.
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How to Use the Quadratic Formula Calculator
- 1Input the coefficients a, b, and c from your quadratic equation (ensure a ≠ 0).
- 2The calculator instantly calculates the discriminant and identifies the nature of the roots.
- 3Review the step-by-step substitution into x = (-b ± √(b² - 4ac)) / (2a).
- 4Copy the factored format, vertex coordinates, and exact roots.
Formula & How It Works
The Quadratic Formula
Derived by completing the square on the general quadratic polynomial, this formula determines the roots x for any quadratic equation.
Worked Examples
Two Distinct Real Roots
Input: a = 1, b = -5, c = 6
Explanation: b² - 4ac = 25 - 24 = 1. x = (5 ± 1) / 2 = 3 and 2.
Complex Conjugate Roots
Input: a = 1, b = 2, c = 5
Explanation: √(-16) = 4i. x = (-2 ± 4i) / 2 = -1 ± 2i.
Frequently Asked Questions
What does a negative discriminant mean?▼
A negative discriminant (b² - 4ac < 0) indicates that the parabola does not cross the x-axis and has two complex (imaginary) solutions.
Can coefficient 'a' be zero?▼
No, if a = 0, the equation reduces to a linear equation (bx + c = 0) rather than a quadratic equation.
Does the calculator find the vertex of the parabola?▼
Yes, the vertex x-coordinate is given by -b / (2a), and the corresponding y-value is reported.
Can I enter decimals or fractions for coefficients?▼
Yes, decimal inputs are supported.
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