Interactive Quadratic Formula Calculator
Calculated result
Enter the known values and select Calculate.
How to use this calculator
- Rewrite the equation in standard form ax² + bx + c = 0.
- Enter zero for a missing b or c term.
- Enter a nonzero value for a.
- Read the discriminant and root type before reviewing the calculated roots.
Formula and method
For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant D = b² − 4ac determines the root type.
Worked example
For x² − 5x + 6 = 0, D = 25 − 24 = 1. The roots are (5 + 1) ÷ 2 = 3 and (5 − 1) ÷ 2 = 2.
Assumptions and result interpretation
- Coefficients are real finite numbers.
- Complex roots are displayed in a + bi form when the discriminant is negative.
- Decimal results are rounded for display while the formula uses full numeric precision.
- The tool solves the equation; it does not factor or graph the polynomial.
Understanding the result
A positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative discriminant gives two complex-conjugate roots.
Common mistakes
- Leaving a missing coefficient blank instead of entering zero
- Forgetting to move every term to one side
- Dropping the ± sign
- Dividing only the square-root term by 2a
Quadratic formula
x = (−b ± √(b² − 4ac)) / 2a. Enter a nonzero value for a and include zero for a missing b or c term.
What the discriminant means
- D > 0: two distinct real roots.
- D = 0: one repeated real root.
- D < 0: two complex-conjugate roots.
Quadratic formula example
For x² − 5x + 6 = 0, the discriminant is 1 and the roots are (5 ± 1) ÷ 2, giving x = 2 and x = 3.
Methodology sources
Use this result as a calculation aid. Confirm input definitions, required precision, and any governing policy or professional standard separately.