If the quantity under the square root ( the discriminant ) is negative, then the ray does not intersect the sphere.

If the discriminant b < sup > 2 </ sup > − 4ac < 0, there are no solutions.

If the discriminant is equal to 0, then there is a single solution, where the line is tangent to the circle.

If the discriminant Δ = g < sub > 2 </ sub >< sup > 3 </ sup > − 27g < sub > 3 </ sub >< sup > 2 </ sup > is not zero, no two of these roots are equal.

If the quadratic function ( 2 ) has a non-negative discriminant (), it has real roots a < sub > 1 </ sub > and a < sub > 2 </ sub > ( not necessarily distinct ):

If a form's discriminant is a fundamental discriminant, then the form is primitive.

If Δ denotes the relative discriminant of L / K, the Artin symbol ( or Artin map, or ( global ) reciprocity map ) of L / K is defined on the group of prime-to-Δ fractional ideals,, by linearity:

If the quadratic polynomial is monic then the roots are quadratic integers.

If the areas of the two parallel faces are A < sub > 1 </ sub > and A < sub > 3 </ sub >, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A < sub > 2 </ sub >, and the height ( the distance between the two parallel faces ) is h, then the volume of the prismatoid is given by ( This formula follows immediately by integrating the area parallel to the two planes of vertices by Simpson's rule, since that rule is exact for integration of polynomials of degree up to 3, and in this case the area is at most a quadratic in the height.

If the matrix is positive semidefinite matrix, then is a convex function: In this case the quadratic program has a global minimizer if there exists some feasible vector ( satisfying the constraints ) and if is bounded below on the feasible region.

where x is the variable, and has a constant term of c. If c = 0, then the constant term will not actually appear when the quadratic is written.

: If then is a quadratic nonresidue

: If is a quadratic residue then

: If then may or may not be a quadratic residue.

* If d is a square free integer then the extension K = Q (√) is a quadratic field of rational numbers.

If the cost function involves quadratic inequalities it is called the quadratic assignment problem.

If the quadratic function is set equal to zero, then the result is a quadratic equation.

If there exists a non-zero v in V such that, the quadratic form Q is isotropic, otherwise it is anisotropic.

If a is zero but one of the other coefficients is non-zero, the equation is classified as either a quartic equation, cubic equation, quadratic equation or linear equation.

If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO ( V, Q ) is the group of linear transformations T of V such that for all x in V there exists a scalar λ such that

If the target is t, then a quadratic loss function is

If F is a quadratic form in n variables, then the theta function associated with F is

If d > 0 the corresponding quadratic field is called a real quadratic field, and for d < 0 an imaginary quadratic field or complex quadratic field, corresponding to whether its archimedean embeddings are real or complex.

If one takes the other cyclotomic fields, they have Galois groups with extra 2-torsion, and so contain at least three quadratic fields.

If d is a square-free integer and K = Q ( d < sup > 1 / 2 </ sup >) is the corresponding quadratic field, then an integral basis of O < sub > K </ sub > is given by ( 1, ( 1 + d < sup > 1 / 2 </ sup >)/ 2 ) if d ≡ 1 ( mod 4 ) and by ( 1, d < sup > 1 / 2 </ sup >) if d ≡ 2 or 3 ( mod 4 ).

If the polynomial is a polynomial in one variable, it determines a quadratic function in one variable.

If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.

If a union cannot perform this function, then collective bargaining is being palmed off by organizers as a gigantic fraud.

If the Greek letter is used, it is assumed to be a Fourier transform of another function,

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

If the function R is well-defined, its value must lie in the range, with 1 indicating perfect correlation and − 1 indicating perfect anti-correlation.

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

If f is not a function, but is instead a partial function, it is called a partial operation.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

If your side has two aces and a void, then you are not at risk of losing the first two tricks, so long as ( a ) your void is useful ( i. e., does not duplicate the function of an ace that your side holds ) and ( b ) you are not vulnerable to the loss of the first two tricks in the fourth suit ( because, for instance, one of the partnership hands holds a singleton in that suit or the protected king, giving your side second round control ).

If evolutionary processes are blind to the difference between function F being performed by conscious organism O and non-conscious organism O *, it is unclear what adaptive advantage consciousness could provide.

If the wave function is regarded as ontologically real, and collapse is entirely rejected, a many worlds theory results.

If wave function collapse is regarded as ontologically real as well, an objective collapse theory is obtained.

If we take the simple valence bond structure and mix in all possible covalent and ionic structures arising from a particular set of atomic orbitals, we reach what is called the full configuration interaction wave function.

If we take the simple molecular orbital description of the ground state and combine that function with the functions describing all possible excited states using unoccupied orbitals arising from the same set of atomic orbitals, we also reach the full configuration interaction wavefunction.