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The 15 pages that link to Discriminant (polynomial), each with the reason it gives.

• Conic sectionRelated: For a quadratic curve, its corresponding coefficient test helps identify the conic type.

• Quadratic equationBroader topic: Its sign predicts whether the quadratic has two, one, or no real roots.

• Invariant theoryBroader topic: Discriminants are classical invariants used to detect special forms and singular cases.

• Completing the squareRelated: The completed-square equation exposes the same quantity that classifies the roots.

• Multiplicity of a rootRelated: It detects multiplicity without explicitly finding the roots.

• Quadratic formulaBroader topic: Its sign tells whether the formula yields two real roots, one repeated root, or complex roots.

• Cubic equationRelated: The cubic discriminant reveals whether roots repeat and how many are real.

• Vieta's formulasRelated: It summarizes root behavior through coefficients, complementing Vieta's descriptions of root sums and products.

• Branch pointRelated: For algebraic functions, its zeros commonly mark values where branches collide.

• Homogeneous polynomialRelated: Discriminants are often homogeneous polynomials in the coefficients they test.

• Quadratic functionRelated: Its sign tells whether the parabola crosses the x-axis twice, touches it, or misses it.

• Algebraic functionRelated: Its zeros locate inputs where distinct algebraic branches can merge.

• Trinomial (algebra)Related: It predicts whether a quadratic trinomial has two, one, or no real zeros.

• Sign (mathematics)Related: Its sign distinguishes the real-root cases of a quadratic equation.

• Casas-Alvero conjectureRelated: The first derivative condition forces a repeated root, detectable through the discriminant.