For example, consider the quadratic polynomial

For r =-1, 1 and 2 we have the harmonic, the arithmetic and the quadratic means respectively.

Thus Legendre's contribution lay in introducing a convenient notation that recorded quadratic residuosity of a mod p. For the sake of comparison, Gauss used the notation, according to whether a is a residue or a non-residue modulo p.

For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the solvability of quadratic equations

For instance, one could define a general quadratic function by defining

For a given quadratic form, several different spaces of spinors with extra properties may exist.

For example, the discriminant of the quadratic polynomial

For example, in the quadratic polynomial

For example, consider the venerable quadratic equation:

For example, in a pseudo-Euclidean space one has the use of a quadratic form:

Much attention has been given to the special case that the function f is a polynomial ; there exist root-finding algorithms exploiting the polynomial nature of f. For a univariate polynomial of degree less than five, there are closed form solutions such as the quadratic formula which produce all roots.

For a free particle, the dispersion relation is a quadratic, and so the effective mass would be constant ( and equal to the real mass ).

For a given n a list of the quadratic residues modulo n may be obtained by simply squaring the numbers 0, 1, …, n − 1.

For this reason some authors add to the definition that a quadratic residue q must not only be a square but must also be relatively prime to the modulus n.

For example, if p ≡ 1 ( mod 8 ), ( mod 12 ), ( mod 5 ) and ( mod 28 ), then by the law of quadratic reciprocity 2, 3, 5, and 7 will all be residues modulo p, and thus all numbers 1 – 10 will be.

For such a function, a smooth quadratic interpolant like the one used in Simpson's rule will give good results.

We can use this fact to prove part of a famous result: for any prime p such that p ≡ 1 ( mod 4 ) the number (− 1 ) is a square ( quadratic residue ) mod p. For suppose p = 4k + 1 for some integer k. Then we can take m = 2k above, and we conclude that

For some small fields, such as the field of rational numbers or its quadratic imaginary extensions there is a more detailed theory which provides more information.

For instance, a coherent state describes the oscillating motion of the particle in a quadratic potential well ( for an early reference, see e. g.

For an equation involving a non-monic quadratic, the first step to solving them is to divide through by the coefficient of x < sup > 2 </ sup >.

For the case of the quadratic kernel, we have:

For example, quadratic fields K of degree 2 over Q are either real ( and then totally real ), or complex, depending on whether the square root of a positive or negative number is adjoined to Q.

Later Helion wrote of this phase: `` For years I built for myself a subtle instrument of relationships -- colors and forms without a name.

For this step the computer memory is separated into three regions: cells in the W-region are used for storage of the forms in the text-form list ; ;

For example, the inflected forms of a word can be represented, insofar as regular inflection allows, by a stem and a set of endings to be attached.

For some forms of philosophy, this very division between the empirical and the rational becomes a sign of the metaphysical superiority of the latter.

For the illusion of depth created by the contrast between the affixed material and everything else gives way immediately to an illusion of forms in bas-relief, which gives way in turn, and with equal immediacy, to an illusion that seems to contain both -- or neither.

For instance, according to Classical Arabic and Modern Standard Arabic, the Arabic root ( to sacrifice ) can be derived the forms ( he sacrificed ), ( you ( masculine singular ) sacrificed ), ( he slaughtered ), ( he slaughters ), and ( slaughterhouse ).

For substances which have two ( or more ) stable crystalline forms, such as diamond and graphite for carbon, there is a kind of " chemical degeneracy ".

For instance member nations of the Commonwealth where English is not spoken natively, such as India, often closely follow British English forms, while many American English usages are followed in other countries which have been historically influenced by the United States, such as the Philippines.

For other words ( such as dreamed, leaned, and learned ) the regular forms are somewhat more common.

For example, the parent function y = 1 / x has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant.

For example, water ice is ordinarily found in the hexagonal form Ice I < sub > h </ sub >, but can also exist as the cubic Ice I < sub > c </ sub >, the rhombohedral ice II, and many other forms.

For example, diamond and graphite are two crystalline forms of carbon, while amorphous carbon is a noncrystalline form.

For a country-by-country listing of officially recognized forms of business organization, see Types of business entity.

For example, ancient Hindu religious texts list clairvoyance amongst other forms of ' clear ' experiencing, as siddhis, or ' perfections ', skills that are yielded through appropriate meditation and personal discipline.

For dance forms without an association structure such as Salsa or Tango Argentino they may not have formal training.

For example, a form input element could be accessed as either " document. formName. inputName " or " document. forms. elements.

For example, Duncan Kennedy, in explicit reference to semiotics and deconstruction procedures, maintains that various legal doctrines are constructed around the binary pairs of opposed concepts, each of which with a claim upon intuitive and formal forms of reasoning that must be made explicit, not only in their meaning but also its relative value, and criticized.

For example, using क ka, the following forms can be derived: क े ke, क ु ku, क ी kī, क ा kā, etc.

* For vowels as an independent syllable ( in writing, unattached to a consonant ), either at the beginning of a word or ( in Hindi ) after another vowel, there are full-letter forms.

For any positive integer n, the set of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted R < sup > n </ sup > and sometimes called real coordinate space.

For the inhomogeneous anisotropic media these two forms of the diffusion equation should be combined in

For some events they can be made in extended forms without a frame at all, but only a long parallel rank of laser beams yet are still called laser harps.

For example, Peter Hain interprets libertarian socialism as minarchist rather than anarchist, favoring radical decentralization of power without going as far as the complete abolition of the state and libertarian socialist Noam Chomsky supports dismantling all forms of unjustified social or economic power, while also emphasizing that state intervention should be supported as a temporary protection while oppressive structures remain in existence.

For example, Dalton knew that the element carbon forms two oxides by combining with oxygen in different proportions.