Discriminant Calculator

Quadratic (Degree 2)

\( ax^2 + bx + c = 0 \)

\( \Delta = b^2 - 4ac \)

• If \(\Delta > 0\), there are 2 distinct real roots.
• If \(\Delta = 0\), there is 1 real root (a repeated root).
• If \(\Delta < 0\), there are 2 complex conjugate roots.

Cubic (Degree 3)

\( ax^3 + bx^2 + cx + d = 0 \)

\( \Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2 \)

• If \(\Delta > 0\), there are 3 distinct real roots.
• If \(\Delta = 0\), there is a repeated root.
• If \(\Delta < 0\), there is 1 real root and 2 complex conjugate roots.

Higher Degrees

The formulas for quartic (degree 4) and quintic (degree 5) discriminants are extremely long and complex but follow similar principles to determine the nature of the polynomial's roots.