noun
definition
A quadratic polynomial, function or equation.
adjective
definition
Square-shaped
definition
Of a polynomial, involving the second power (square) of a variable but no higher powers, as ax^2 + bx + c.
definition
Of an equation, of the form ax^2 + bx + c = 0.
definition
Of a function, of the form y = ax^2 + bx + c .
Examples of quadratic in a Sentence
The next case is that in which u is a quadratic function of x, i.e.
He solved quadratic equations both geometrically and algebraically, and also equations of the form x 2 "+ax n +b=o; he also proved certain relations between the sum of the first n natural numbers, and the sums of their squares and cubes.
The principle underlying this expression is probably to be found in the fact that it measured the limits of their attainments in algebra, for they were unable to solve equations of a higher degree than the quadratic or square.
For the quadratic it is the discriminant (ab) 2 and for ax2 the cubic the quadratic covariant (ab) 2 axbx.
For example, take the ternary quadratic (aixl+a2x2+a3x3) 2 =a2x, or in real form axi +bx2+cx3+2fx 2 x 3+ 2gx 3 x 1 +2hx i x 2.
Similarly, For A Linear And A Quadratic, P= I, Q= 2, And The Reduced Form Is Found To Be 1 A2B2Z2 1 Az.
Of the quadratic axe+2bxy+cy2, he discovered the two invariants ac-b 2, a-2b cos w+c, and it may be verified that, if the transformed of the quadratic be AX2=2BXY+CY2, sin w 2 AC -B 2 =) (ac-b2), sin w A-2B cos w'+C = (sin w'1 2(a - 2bcosw+c).
Henry Thomas Colebrooke, one of the earliest modern investigators of Hindu science, presumes that the treatise of Aryabhatta extended to determinate quadratic equations, indeterminate equations of the first degree, and probably of the second.
A notable improvement on the ideas of Diophantus is to be found in the fact that the Hindus recognized the existence of two roots of a quadratic equation, but the negative roots were considered to be inadequate, since no interpretation could be found for them.
These frozen metals in general form compact masses consisting of aggregates of crystals belonging to the regular or rhombic or (more rarely) the quadratic system.
Conversely, if the kinetic energy T is expressed as a quadratic function of x, x x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.
The ordinary hydrated variety forms quadratic crystals and behaves as a strong base.
It follows from §§ 48 and 51 that, if V is a solid figure extending from a plane K to a parallel plane L, and if the area of every cross-section parallel to these planes is a quadratic function of the distance of the section from a fixed plane parallel to them, Simpson's formula may be applied to find the volume of the solid.
In the case of the sphere, for instance, whose radius is R, the area of the section at distance x from the centre is lr(R 2 -x 2), which is a quadratic function of x; the values of So, Si, and S2 are respectively o, 7rR 2, and o, and the volume is therefore s.
By drawing Ac and Ad parallel to BC and BD, so as to meet the plane through CD in c and d, and producing QP and RS to meet Ac and Ad in q and r, we see that the area of Pqrs is (x/h - x 2 /h 2) X area of cCDd; this also is a quadratic function of x.
In the case, therefore, of any solid whose cross-section at distance x from one end is a quadratic function of x, the position of the crosssection through the centroid is to be found by determining the position of the centre of gravity of particles of masses proportional to So, S2, and 4S 1, placed at the extremities and the middle of a line drawn from one end of the solid to the other.
His mathematical writings, which account for some forty entries in the Royal Society's catalogue of scientific papers, cover a wide range of subjects, such" s the theory of probabilities, quadratic forms, theory of integrals, gearings, the construction of geographical maps, &c. He also published a Traite de la theorie des nombres.
It crystallizes in quadratic prisms.
To Legendre is due the theorem known as the law of quadratic reciprocity, the most important general result in the science of numbers which has been discovered since the time of P. de Fermat, and which was called by Gauss the " gem of arithmetic."
It may be obtained crystallized in the quadratic system by melting in a sealed tube containing hydrogen, allowed to cool partially, and then pouring off the still liquid portion by inverting the tube.
Now these integrations are quite intractable, even for a very simple mathematical assumption of the function f(v), say the quadratic or cubic law, f(v) = v 2 /k or v3/k.
It may be obtained crystallized in quadratic octahedra of a greenish-blue colour, by melting in a sealed tube containing an inert gas, and inverting the tube when the metal has partially solidified.
The three commonest means are the arithmetical, geometrical, and harmonic; of less importance are the contraharmonical, arithmetico-geometrical, and quadratic.
The quadratic mean of n quantities is the square root of the arithmetical mean of their squares.
In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian parabola; a 2 x = y 3 is the cubic parabola, a 3 x = y4 is the biquadratic parabola; semi parabolas have the general equation ax n-1 = yn, thus ax e = y 3 is the semicubical parabola and ax 3 = y 4 the semibiquadratic parabola.
In his Treatise of Algebra (1685) he distinctly proposes to construct the imaginary roots of a quadratic equation by going out of the line on which the roots, if real, would have been constructed.
Of the phosphotungstic acids the most important is phosphoduodecitungstic acid, H 3 PW, 2040 nH 2 O, obtained in quadratic pyramids by crystallizing mixed solutions of orthophosphoric and metatungstic acids.
Silicotungstic acid is obtained as quadratic pyramids from its mercurous salt which is prepared from mercurous nitrate and the salt formed on boiling gelatinous silicic acid with a polytungstate of an alkali metal.
We proceed to the theory of the plane, axial and polar quadratic moments of the system.
Another type of quadratic moment is supplied by the deviationmoments, or products of inertia of a distribution of matter.
The quadratic moment,s with respect to different planes through a fixed point 0 are related to one another as follows.
Evidently the quadratic moment for a variable plane through 0 will have a stationary value when, and only when, the plane coincides with a principal plane of (26).
The distance between the planes of and of will be of the second order of small quantities, and the quadratic moments with respect to of and co will therefore be equal, to the first order.
The directions of these axes are determined by the property (24), and therefore coincide with those of the principal axes of inertia at 0, as already defined in connection with the theory of plane quadratic moments.
If we replace the mass of each particle by its moment, as thus found, we can in like manner obtain the quadratic moment of the system with respect to the line.
The quadratic moment of the first particle will then be represented by twice the area FIG.
The quadratic moment of the whole system is therefore represented by twice the area AHEDCBA.
Since a quadratic moment is essentrally positive, the various areas are to taken positive in all cases.
If some of the particles lie on one side of p and some on the other, the quadratic moment of each set may be found, and the results added.
If the quadratic (38) has a negative root, the trigonometrical functions in (36) are to be replaced by real exponentials, and the position x=o, y=o is unstable.
Since T is a homogeneous quadratic function.
The quadratic expression for T is essentially positive, and the same holds with regard to V in virtue of the assumed stability.
The value of such a fraction is the positive root of a quadratic equation whose coefficients are real and of which one root is negative.
Since the fraction is infinite it cannot be commensurable and therefore its value is a quadratic surd number.
Conversely every positive quadratic surd number, when expressed as a simple continued fraction, will give rise to a recurring fraction.
The second case illustrates a feature of the recurring continued fraction which represents a complete quadratic surd.
A saturated solution of the hydroxide deposits on cooling a hydrated form Ba(OH) 2.8H 2 0, as colourless quadratic prisms, which on exposure to air lose seven molecules of water of crystallization.
Silver fluoride, AgF, is obtained as quadratic octahedra, with one molecule of water, by dissolving the oxide or carbonate in hydrofluoric acid.
Consider the general quadratic equation ax 2 + bx + c = 0 where a 0.
A quadratic average stress failure criterion is suggested to predict delamination and the interlayer at which it occurs.