Education
Quadratic Discriminant Calculator
Use the discriminant to predict the type of solutions a quadratic equation has before solving the full quadratic formula.
What the discriminant tells you
The discriminant is the part of the quadratic formula under the square root: b² - 4ac. It tells you what type of solutions a quadratic equation has before you fully solve it. OpenStax uses the discriminant for exactly this purpose in its quadratic formula lesson.
This is useful because the sign of the discriminant is a fast diagnostic. A positive discriminant means the graph crosses the x-axis twice. A zero discriminant means it touches the x-axis once at the vertex. A negative discriminant means there are no real x-intercepts.
Discriminant decision table
| Discriminant | Solutions | Graph meaning |
|---|---|---|
| D > 0 | Two real solutions | Two x-intercepts |
| D = 0 | One repeated real solution | Touches x-axis once |
| D < 0 | Two complex solutions | No real x-intercepts |
Worked example
x² - 5x + 6 = 0\n\nD = b² - 4ac\nD = (-5)² - 4(1)(6)\nD = 25 - 24 = 1\n\nSince D > 0, the equation has two distinct real solutions.
The roots are x = 2 and x = 3, which match the factorization (x - 2)(x - 3).
Common questions
-
The discriminant is b² - 4ac, the expression under the square root in the quadratic formula. It is used to determine the number and type of solutions of a quadratic equation without necessarily solving the equation completely.
-
A positive discriminant means the quadratic has two distinct real solutions. On the graph, the parabola crosses the x-axis at two different points. If the discriminant is a perfect square, the real solutions are rational; if it is positive but not a perfect square, the solutions are irrational.
-
A zero discriminant means the quadratic has one repeated real solution. The parabola touches the x-axis at exactly one point, which is also the vertex. Algebraically, the quadratic is a perfect square trinomial or behaves like one after simplification.
-
A negative discriminant means the quadratic has no real solutions, because the quadratic formula would require the square root of a negative number. The equation still has two complex solutions, and the graph does not cross the x-axis.
-
No. The discriminant is only one part of the quadratic formula. The full formula is x = (-b ± √(b² - 4ac))/(2a). The discriminant helps classify the solutions before or while using the full formula.