Quadratic Formula Calculator
Solve ax² + bx + c = 0 and find the discriminant, real or complex roots, vertex, axis of symmetry, intercepts, and root checks.
Quadratic formula and discriminant
For ax² + bx + c = 0, with a ≠ 0, the quadratic formula is x = (−b ± √(b² − 4ac)) ÷ (2a). The expression D = b² − 4ac is the discriminant.
- If D > 0, the equation has two distinct real roots.
- If D = 0, it has one repeated real root.
- If D < 0, it has two complex conjugate roots.
The vertex is (−b/(2a), −D/(4a)), and its vertical axis of symmetry is x = −b/(2a). By Viète’s formulas, the roots have sum −b/a and product c/a.
Checked examples
| Equation | Discriminant | Roots | Vertex |
|---|---|---|---|
| x² − 3x + 2 = 0 | 1 | 1, 2 | (1.5, −0.25) |
| x² + 2x + 1 = 0 | 0 | −1 repeated | (−1, 0) |
| x² + 1 = 0 | −4 | i, −i | (0, 1) |
| 2x² + 5x − 3 = 0 | 49 | −3, 0.5 | (−1.25, −6.125) |
| −x² + 4x − 4 = 0 | 0 | 2 repeated | (2, 0) |
| x² − 4 = 0 | 16 | −2, 2 | (0, −4) |
| 0.5x² − x − 1.5 = 0 | 4 | −1, 3 | (1, −2) |
| x² + 100000000x + 1 = 0 | ≈10¹⁶ | ≈−100000000, −0.00000001 | (−50000000, ≈−2.5×10¹⁵) |
How to use the calculator
- Enter the coefficients a, b, and c from ax² + bx + c = 0.
- Keep the signs with their coefficients; for x² − 3x + 2, enter 1, −3, and 2.
- Select Solve quadratic equation.
- Review the roots, discriminant, vertex, symmetry axis, intercept, direction, and root checks.
Limits and numerical method
- Each coefficient must be finite and no larger than 1e50 in magnitude.
- Coefficient a cannot be zero. A zero value would produce a linear or constant equation.
- Commas are rejected to avoid ambiguity between decimal and thousands separators.
- Real roots use a numerically stable form of the quadratic formula to reduce loss of precision when one root is much smaller than the other.
- Displayed values are rounded to 12 significant digits; calculations use JavaScript double-precision arithmetic.
- The calculator solves one-variable quadratic equations; it does not fit a quadratic model to data or draw a graph.
Frequently asked questions
What does a negative discriminant mean?
It means there are no real x-intercepts. The two solutions contain the imaginary unit i and form a complex-conjugate pair.
Why are the sum and product of roots included?
They provide a quick independent check: the sum should equal −b/a and the product should equal c/a.
Can this solve an equation with a missing term?
Yes. Enter zero for the missing coefficient. For x² − 4 = 0, use a=1, b=0, and c=−4.
Does the order of the roots matter?
No. For two real solutions, the calculator lists the smaller root first. Both values solve the same equation.
Related calculators
Use the Square Root Calculator to evaluate and simplify radicals, the Scientific Notation Calculator for very large and small values, the Fraction Calculator for exact rational arithmetic, or the GCF and LCM Calculator for integer factors and multiples.