Quadratic Formula Calculator · Practical guide
What Does the Discriminant Tell You?
D = b² − 4ac predicts the roots before you solve: positive means two real roots, zero means one, negative means complex.
At a glance
Quick answer
The discriminant, D = b² − 4ac, tells you the number and type of roots a quadratic equation has — before you finish solving it. D > 0 means two distinct real roots, D = 0 means one repeated real root, and D < 0 means two complex conjugate roots. It is the part under the square root in the quadratic formula, which is why its sign controls everything.
Interactive tool
Quadratic Formula Calculator
Enter your own values, then compare the result with the examples below.
Solution
Step-by-Step:
The Three Cases with Examples
| Discriminant | Roots | Example | Graph behavior |
|---|---|---|---|
| D > 0, perfect square | Two rational roots | x² − 5x + 6 = 0 → x = 2, 3 | Crosses x-axis twice |
| D > 0, not a square | Two irrational roots | x² − 2 = 0 → x = ±1.4142 | Crosses x-axis twice |
| D = 0 | One repeated root | x² − 6x + 9 = 0 → x = 3 | Touches x-axis at vertex |
| D < 0 | Two complex roots | x² + 2x + 5 = 0 → x = −1 ± 2i | Never touches x-axis |
Why It Works
The quadratic formula is x = (−b ± √D) / 2a. When D is positive, √D is a real number and the ± splits the answer in two. When D = 0, the ± adds and subtracts zero, collapsing to the single root x = −b / 2a. When D is negative, √D introduces the imaginary unit i, and the two roots become conjugates p ± qi.
The Perfect-Square Bonus
When D is a positive perfect square — 1, 4, 9, 16, 25 — the roots come out rational, which means the equation could have been factored over the integers. For x² − 5x + 6 = 0, D = 1 = 1², and indeed the roots 2 and 3 are whole numbers. For x² − 2 = 0, D = 8 is not a perfect square, so the roots ±√2 stay irrational. The calculator above reports D to four decimal places alongside the roots, so you can spot the case at a glance.
Frequently Asked Questions
Can the discriminant be a fraction or decimal?
Yes. If a, b, or c are decimals, D can be too — for 0.5x² − x + 0.25 = 0, D = 1 − 0.5 = 0.5, and the roots are real but irrational since 0.5 is not a perfect square.
Does a negative discriminant mean no solution?
It means no real solution. The equation still has two complex roots, like −1 ± 2i for x² + 2x + 5 = 0, and the parabola simply never crosses the x-axis.
How does the discriminant relate to the graph?
It counts x-intercepts: D > 0 means the parabola crosses the axis twice, D = 0 means it just touches at the vertex, and D < 0 means it floats entirely above or below the axis.
What is the discriminant of 3x² + 2x + 1?
D = 2² − 4(3)(1) = 4 − 12 = −8. Because it is negative, the roots are complex: x = (−2 ± √8 i) / 6, or about −0.3333 ± 0.4714i.