Discriminant Calculator

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Choose quadratic for degree-2, cubic for degree-3. The discriminant formula and its interpretation change with degree.
The coefficient of the highest-degree term. Must be non-zero for the polynomial to have the chosen degree.
For a quadratic this is the x coefficient; for a cubic it is the x² coefficient.
For a quadratic this is the constant term; for a cubic it is the x coefficient.
When on, the calculator solves the quadratic formula to show the actual root values in addition to the discriminant.
Discriminant (Δ)Real roots exist
1
Nature of the rootsTwo distinct real roots
Number of real rootsTwo
Root typeTwo rational (exact)
Root x₁3
Root x₂2
Axis of symmetry (x =)2.5
Vertex (h, k)(2.5, -0.25)
Parabola opensUpward (minimum at vertex)
1
Complex roots<0One root0-0.0001Two real roots0.0001+

Delta = 1 > 0: the parabola crosses the x-axis at two distinct points.

  • Root type: Two rational (exact). Because the discriminant is a perfect square, the roots are rational numbers.
  • The parabola opens upward with a minimum at the vertex (2.5, -0.25).
  • Axis of symmetry: x = 2.5.

Next stepThe two roots are x₁ = 3 and x₂ = 2.

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