Quadratic formula calculator for real and complex roots
Enter a, b and c for ax²+bx+c=0. HitEquals reports both roots, discriminant, root type, vertex, symmetry axis and a substitution check.
Inputs used by this calculator
Quadratic Formula Calculator uses the fields below. The starting values reproduce the page example; replace them with values from the same units and time period before using the result.
| Input | Starting example | Accepted value | How it is used |
|---|---|---|---|
| Coefficient a | 1 | -1000000000 to 1000000000 | Supplies the coefficient a quantity used by the published equation or result breakdown. |
| Coefficient b | -5 | -1000000000 to 1000000000 | Supplies the coefficient b quantity used by the published equation or result breakdown. |
| Coefficient c | 6 | -1000000000 to 1000000000 | Supplies the coefficient c quantity used by the published equation or result breakdown. |
Standard quadratic form
A quadratic equation has ax²+bx+c=0 with a not equal to zero. If a were zero, the equation would be linear.
Quadratic formula
The formula solves every quadratic in standard form. Its square-root term contains the discriminant.
x=(−b±√D)/(2a)D=b²−4acWorked factorable example
For a=1, b=−5 and c=6, D=25−24=1. The two signs produce (5−1)/2=2 and (5+1)/2=3.
Reading the discriminant
When D is positive there are two distinct real roots; when zero the same real root repeats; when negative the roots contain imaginary parts.
Vertex and symmetry
The vertex x-coordinate and axis of symmetry are −b/(2a). Substituting that x-value into the polynomial gives the vertex y-coordinate.
Numerical precision
Very large or nearly cancelling coefficients can magnify floating-point rounding. Symbolic algebra or higher precision may be required for sensitive work.
Discriminant guide
| D | Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | One repeated real root |
| D < 0 | Two complex conjugate roots |
Frequently asked questions
What if a equals zero?
The equation is linear, so the quadratic formula does not apply.
What does a negative discriminant mean?
The equation has a complex-conjugate root pair.
Can a quadratic have one root?
It can have one repeated real root when the discriminant is zero.
What is the axis of symmetry?
x=−b/(2a).
How can I check a real root?
Substitute it into ax²+bx+c; the result should be near zero.
Why are decimals rounded?
The display rounds while the calculation retains floating-point precision.
What to keep in mind
Coefficient a must be nonzero. Extremely ill-conditioned coefficients can lose decimal precision in standard floating-point arithmetic.
Results are rounded for display in US format, with money in dollars and cents; CSV exports keep full calculation precision. This is a calculation summary, not an official certificate.