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How to use
The quadratic equation calculator solves ax²+bx+c=0 when a≠0. It first evaluates the discriminant to identify the root type, then reports real or complex roots together with the parabola vertex, axis of symmetry and y-intercept.
- Rewrite the equation as ax²+bx+c=0.
- Enter a, b and c; use zero for a missing term, but a itself cannot be zero.
- Calculate and review the discriminant, roots, vertex and symmetry axis.
How it is calculated
Example
Example: x²−5x+6=0 has a=1, b=−5 and c=6. The discriminant is 1, so the roots are 2 and 3. The vertex is (2.5, −0.25) and the symmetry axis is x=2.5.
Important notes
If Δ>0 there are two distinct real roots; Δ=0 gives one repeated real root; Δ<0 gives a complex conjugate pair. For real roots the calculator uses a numerically more stable variant of the quadratic formula to reduce cancellation error in some cases, but browser floating-point precision still applies and this is not a symbolic algebra system.
Worked examples and interpreting results
The equation x²−5x+6=0 has coefficients a=1, b=−5 and c=6. Its discriminant is 1, so it has two distinct real roots, 2 and 3.
| Case | Calculation | Result |
|---|---|---|
| Discriminant | (−5)² − 4 × 1 × 6 | 1 |
| First root | (5 − √1) ÷ 2 | 2 |
| Second root | (5 + √1) ÷ 2 | 3 |
How to check the result
Substitute each root into the original equation: 2²−5×2+6 equals zero. A zero discriminant gives a repeated root; a negative one gives no real roots. If a=0, the equation is linear rather than quadratic.
Frequently asked questions
What if a equals zero?
Then the equation is not quadratic, so this calculator rejects it instead of applying the wrong formula.
What does a negative discriminant mean?
There are no real roots; the solutions form a complex conjugate pair.
Is the vertex the same as a root?
No. The vertex is the parabola’s turning point, while roots are x-values where y=0.