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The quantity b 2- 4 ac, where a,b,c are coefficients of a given quadratic polynomial: ax 2+ bx + c.
More generally, for the polynomial equation a0 xn + a1 xn -1+···+ anx0= 0, a02 n-2times the product of the squares of all the differences of the roots of the equation, taken in pairs.



The discriminant of a polynomial

P(x) = a0xn + a1xn−1 + … + an

is the expression

in which the product is distributed over all possible differences of the roots α1, β2, … , αn of the equation P (x) = 0. The discriminant vanishes if and only if there are equal roots among the roots of the polynomial. The discriminant can be expressed through the coefficients of the polynomial P(x) by representing it in the form of a determinant consisting of these coefficients. Thus, for the second-degree polynomial ax2 + bx + c, the discriminant is b2 − 4ac. For x3 + px + q, the discriminant is −4p3 −27q2. The discriminant differs only by a factor a0 from the resultant R(P, P′) of the polynomial P(x) and its derivative P′(x).