# Quadratic Sentence Examples

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.AdvertisementSimilarly, 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.AdvertisementConversely, 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.AdvertisementIn 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.AdvertisementNow 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.AdvertisementIn 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.The approximations behind the

**quadratic**residue diffuser 's design have been tested.He solved problems such as pairs of simultaneous

**quadratic**equations.An appropriate rescaling casts the system in a normal form, which is universal for models supporting ess through

**quadratic**nonlinearities.The arbitrariness of the choice was underlined later, when the teacher introduced a

**quadratic**graph.Newton strategies additionally require the second partial derivatives, thus building a

**quadratic**internal model.Factorizing quadratics An essential skill in many applications is the ability to factorize

**quadratic**expressions.If you'd like to see how to find the Julia set of the simplest

**quadratic**, have a look at this example.One of the questions asked him to show that a particular

**quadratic**had ' real and distinct roots ' .Thus, the Mesopotamians knew how to solve

**quadratic**equations 4000 years ago, using essentially the same method that we use today.Use this & Gauss ' law of

**quadratic**reciprocity, to show that 75 is a primitive root modulo 65537.The

**quadratic**spline takes longer to render than a linear spline.Knowing the maximum or minimum values and where the graph hits the x-axis (solving the

**quadratic**).For the

**quadratic**aoxi +2a i x i x 2 +a 2 x, we have (i.) ax = 7/1x1+2aixix2-I-7/24, (ii.) xx=xi+xzi (ab) 2 =2(aoa2 - ai), a a = a o+712, _ (v.) (xa)ax= i'?- (a2 - ao)xix2 - aix2.The

**quadratic**equation x 2 +b 2 =o, for instance, has no real root; but we may treat the roots as being +b-' - I, and - b 1, 1 - 1, if -J - i is treated as something which obeys the laws of arithmetic and emerges into reality under the condition 1 1 - I.The theoretical assumptions of Newton and Euler (hypotheses magis mathematicae quam naturales) of a resistance varying as some simple power of the velocity, for instance, as the square or cube of the velocity (the

**quadratic**or cubic law), lead to results of great analytical complexity, and are useful only for provisional extrapolation at high or low velocity, pending further experiment.If you 'd like to see how to find the Julia set of the simplest

**quadratic**, have a look at this example.One of the questions asked him to show that a particular

**quadratic**had ' real and distinct roots '.Solve a

**quadratic**equation using factors Errors Identify sources of errors.Values of 1 or 2 will cause linear or

**quadratic**interpolation to be used.Context

**quadratic**time prominent role in these assets were in canada should.Factorizing Quadratics There is no simple method of factorizing a

**quadratic**expression, but with a little practice it becomes easier.In particular, the

**quadratic**regularizing term discourages sudden changes in surface normal direction across a surface.**Quadratic**residue diffuser In this case, the longest wells in the diffuser were replaced by shorter, active equivalents.Finally, a forward-looking interpretation of the short-run dynamics, assuming

**quadratic**adjustment costs, cannot be rejected by the data.Knowing the maximum or minimum values and where the graph hits the x-axis (solving the

**quadratic**).He was also the author of important papers in which he extended to complex

**quadratic**forms many of Gauss's investigations relating to real**quadratic**forms. After 1864 he devoted himself chiefly to elliptic functions, and numerous papers on this subject were published by him in the Proc. Lond.It forms

**quadratic**prisms, having a violet reflex and insoluble in boiling hydrochloric acid.Among the great variety of problems solved are problems leading to determinate equations of the first degree in one, two, three or four variables, to determinate

**quadratic**equations, and to indeterminate equations of the first degree in one or more variables, which are, however, transformed into determinate equations by arbitrarily assuming a value for one of the required numbers, Diophantus being always satisfied with a rational, even if fractional, result and not requiring a solution in integers.But the bulk of the work consists of problems leading to indeterminate equations of the second degree, and these universally take the form that one or two (and never more) linear or

**quadratic**functions of one variable x are to be made rational square numbers by finding a suitable value for x.His largest work,Trattato generale di numeri e misure, is a comprehensive mathematical treatise, including arithmetic, geometry, mensuration, and algebra as far as

**quadratic**equations (Venice, 1556, 1560).Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences,

**quadratic**residues, prime numbers, particular irrational and transcendental numbers.Arithmetical groups, connected with the theory of

**quadratic**forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and equations, came into existence, and also particular linear and higher transformations connected with analysis and geometry.The expression (ab) 4 properly appertains to a quartic; for a

**quadratic**it may also be written (ab) 2 (cd) 2, and would denote the square of the discriminant to a factor pres.For the cubic (ab) 2 axbx is a covariant because each symbol a, b occurs three times; we can first of all find its real expression as a simultaneous covariant of two cubics, and then, by supposing the two cubics to merge into identity, find the expression of the

**quadratic**covariant, of the single cubic, commonly known as the Hessian.Further, it is convenient to have before us two other

**quadratic**covariants, viz.He also showed that every equation of an even degree must have at least one real

**quadratic**factor, reduced the solution of linear differential equations to definite integrals, and furnished an elegant method by which the linear partial differential equation of the second order might be solved.It includes the properties of numbers; extraction of roots of arithmetical and algebraical quantities, solutions of simple and

**quadratic**equations, and a fairly complete account of surds.Regular crystals expand equally in all directions; rhombic and

**quadratic**expand differently in different directions.Thus if T is expressed as a

**quadratic**function of U, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a**quadratic**function of xi, 'x2, x3, yi, Y2, Y3, U = d, V= dx, ' w= ax dT Q_ dT dT dy 1 dy2 dy The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow X = dt x2 dy +x3 d Y = ..., Z ..., (3) = dt1 -y2?y - '2dx3+x3 ' M =..It crystallizes in

**quadratic**prisms and has a bitter taste.Why, I can do long, complicated

**quadratic**equations in my head quite easily, and it is great fun!The simultaneous system of two

**quadratic**forms ai, ay, say f and 0, consists of six forms, viz.There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a

**quadratic**. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, as we may say, the common harmonic conjugates of the lines and the lines x x .