Quadratic equation

“ The case of most frequent occurrence is that in which the coefficients P, Q are constants; we consider this case in some detail. If θ be a root of the quadratic equation θ2 + θP + Q = 0, it can be at once seen that a particular integral of the differential equation with zero on the right side is y1 = eθx. ”

“ If these problems were in the original text, and that text has the antiquity usually assigned to it, concerning neither of which we are at all certain, then they contain the oldest known quadratic equation. The interrelation of ancient mathematics is seen in two problems in this chapter. One is that of the reed growing 1 foot above the surface in the center of a pond 10 feet square, which just reaches the surface when drawn to the edge of the pond, it being required to find the depth of the water. ”

“ Quadratic equations arose in the Greek investigations in the doctrine of proportion, and although they were presented and solved in a geometrical form, the methods employed have no relation to the generalized conception of algebraic geometry which represents a curve by an equation and vice versa. The simplest quadratic arose in the construction of a mean proportional (x) between two lines (a, b) , or in the construction of a square equal to a given rectangle; for we have the proportion a:x = x:b; i.e. x2 = ab. A more general equation, viz. ”

“ This concept is extended to algebra: since a line, surface and solid are represented by linear, quadratic and cubic equations, and are of one, two and three dimensions; a biquadratic equation has its highest terms of four dimensions, and, in general, an equation in any number of variables which has the greatest sum of the indices of any term equal to n is said to have n dimensions. The “fourth dimension” is a type of non-Euclidean geometry, in which it is conceived that a “solid” has one dimension more than the solids of experience. ”

“ In the majority of cases we only care for one of the roots of the equation, which is distinguished by being very small, and which will be most easily calculated by a small correction to the value which we get by neglecting the quadratic term. ”

“ And A or x is one of the two unequal segments. It may be found thus:
The half of the middle species is (5) , its square is (25) . From it subtract the absolute term AE (21) , and (4) will be the square of half the difference of the segments. The square root of this, (2) , is half the difference. If you add it to half the coefficient (5) , or half the line to be bisected, the longer segment is obtained; if you subtract it, the smaller segment is obtained. I say:
The quadratic equation Aq+ZA=AE receives similar treatment. ”

“ Remarkable is the Hindoo solution of the quadratic equation . With great keenness of intellect they recognised in the special case a fundamental problem in indeterminate quadratics. They solved it by the cyclic method. "It consists," says De Morgan, "in a rule for finding an indefinite number of solutions of (a being an integer which is not a square) , by means of one solution given or found, and of feeling for one solution by making a solution of give a solution of . ”

“ It can be shown, from the fact that η1 is a quadratic polynomial in y1, that no differential equation of the second order can allow more than 8 really independent infinitesimal transformations, and that every homogeneous linear differential equation of the second order allows just 8, being in fact reducible to d2y/dx2 = 0. Since every group of more than two parameters has subgroups of two parameters, a differential equation of the second order allowing a group of more than two parameters can, as a rule, be solved by quadratures. ”

“ I. To find the sum and the product of the roots.
The general quadratic equation is
(1)
Or, (2)
To derive the formula, we have by transposing
Completing the square,
Extracting square root,
Transposing,
Hence,
These two values of x we call roots. For convenience represent them by and
[pg 39]
Hence we have shown that
and
Or, referring to equation (2) above, we have the following rule:
When the coefficient of is unity, the sum of the roots is the coefficient of x with the sign changed; the product of the roots is the independent term.
Examples:
1. ”

“ Arithmetic
Geometry
Physics
[pg 36] QUADRATIC EQUATIONS
1. Define a quadratic equation; a pure quadratic; an affected (or complete) quadratic; an equation in the quadratic form.
2. Solve the pure quadratic
Review the first (or usual) method of completing the square. Solve by it the following:
3.
4.
5.
6.
Review the solution by factoring. Solve by it the following:
7.
8.
9.
10.
Solve, by factoring, these equations, which are not quadratics:
11.
12.
13.
Review the solution by formula. Solve by it the following:
14. ”

“ When a biquadratic equation has all its terms, its resolution may be always reduced to that of a cubic equation. There are various methods by which such a reduction may be effected. The following was first given by Leonhard Euler in the Petersburg Commentaries, and afterwards explained more fully in his Elements of Algebra. ”

“ From this equation and the n − 1 equations Πiω = 0, Pσω = 0, we can determine ω by a quadrature only. Hence can be deduced a much more general result, that if the group of r parameters be integrable, the complete system can be entirety solved by quadratures; it is only necessary to introduce the solution found by the first quadrature as an independent variable, whereby we obtain a complete system of n − r equations in n − 1 variables, subject to an integrable group of r − 1 parameters, and to continue this process. We give some examples of the application of the theorem. ”

“ If, in the case of two variables,
then the quadratic is one-signed unless it vanishes, but the value of u is not necessarily a maximum or minimum, and the terms of the third and possibly fourth order must be taken account of.
Take for instance the function u = x2 − xy2 + y2. Here the values x = 0, y = 0 satisfy the equations δu/δx = 0, δu/δy = 0, so that zero is a critical value of u, but it is neither a maximum nor a minimum although the terms of the second order are (δx) 2, and are never negative. ”

“ The enumeration of all possible cases is thus reduced to a simple question in combinatorial analysis, and the actual study of any particular case is much facilitated by a useful rule of Klein’s for writing down in a simple form two quadratics belonging to a given class—one of which, of course, represents the equation connecting line coordinates, and the other the equation of the complex. The general complex is naturally [111111] ; the complex of tangents to a quadric is [ (111) , (111) ] and that of lines meeting a conic is [ (222) ] . ”

“ Or that raising Resolvends out of assumed roots, those should make a store from whence to derive the roots of the Resolvend offered.
Or making quadratick Equations out of the dioristick and base limits, those might be interpoled, by aid of a Table of figurate numbers, or otherwise thereby, as in quadratick Equations to attain two roots of a biquadratick at once. which if performed the greatest difficulties are overcome, and why should not this seem probable, in regard the Curve or Locus, be the Equation what it will, makes indented porches. ”

“ Any angle is easily trisected by means of certain higher curves, such as the conchoid of Nicomedes (ca. 180 B.C.) , the quadratrix of Hippias of Elis (ca. 420 B.C.) , or the spiral of Archimedes (ca. 250 B.C.) . But since this problem, stated algebraically, requires the solution of a cubic equation, and this involves, geometrically, finding three points, we cannot solve the problem by means of straight lines and circles alone. In other words, the trisection of any angle, by the use of the straightedge and compasses alone, is impossible. Special angles may however be trisected. ”

“ Suppose, for instance, that the equations of a line (depending on 719 the variable parameter θ) are x/a + y/c = θ (1 + y/b) , x/a − z/c = (1/θ) (1 − y/b) ; then, eliminating θ we have x2/a2 − z2/c2 = 1 − y2/b2, or say, x2/a2 + y2/b2 − z2/c2 = 1, the equation of a quadric surface, afterwards called the hyperboloid of one sheet; this surface is consequently a scroll. It is to be remarked that we have upon the surface a second singly infinite series of lines ”

“ The expression a + bx + cx2 may be transformed into a square as often as it can be resolved into two parts, one of which is a complete square, and the other a product of two simple factors; for then it has this form, p2 + qr, where p, q and r are quantities which contain no power of x higher than the first. Let us assume √ (p2 + qr) = p + mq; thus we have p2 + qr = p2 + 2mpq + m2q2 and r = 2mp + m2q, and as this equation involves only the first power of x, we may by proper reduction obtain from it rational values of x and y, as in the three foregoing cases. ”

“ Some problems may be solved by algebra; thus if y = ƒ (x) ÷ φ (x) , where ƒ (x) and φ (x) are polynomials in x, the limits to the values of yφ may be found from the consideration that the equation yφ (x) − ƒ (x) = 0 must have real roots. This is a useful method in the case in which φ (x) and ƒ (x) are quadratics, but scarcely ever in any other case. The problem of finding the maximum product of n positive quantities whose sum is given may also be found, algebraically, thus. ”

“ If we except those cases in which the problem has no definite meaning because the data are such that the integral must be infinite, it is evident that a minimum must always exist, and (on account of the quadratic form of the integral) that it is unique. ”

“ By triumphs must be understood the successes with which the calculus has been applied to the solution of problems otherwise intractable. Often in the consideration of physical relations one is able to build up an expression for the law governing the interaction of the parts or of the forces that govern them, such expression being naturally in the form of a differential equation, that is an equation containing differential coefficients with or without other algebraic quantities. ”

“ Besides the admirable facility which is given by the transcendental analysis for the investigation of the mathematical laws of all phenomena, a second fundamental and inherent property, perhaps as important as the first, is the extreme generality of the differential formulas, which express in a single equation each determinate phenomenon, however varied the subjects in relation to which it is considered. Thus we see, in the preceding examples, that a single differential equation gives the tangents of all curves, another their rectifications, a third their quadratures ”

“ Thus, for example, in the case of rectifications, it will be first necessary to calculate dy, or dy and dz, by differentiating the equation or the equations of each curve proposed; after eliminating these expressions, the general differential formula above enunciated will then contain only ds and dx; having arrived at this point, the elimination of the infinitesimals can be completed only by the integral calculus. ”

“ But if the group of three parameters be simple, this result must be replaced by the statement that the integration is reducible to quadratures and that of a so-called Riccati equation of the first order, of the form dy/dx = A + By + Cy2, where A, B, C are functions of x. ”

“ But from Algebra we know that the quadratic expression
always has the same sign as (or ) when . Applying this to (F) , and we see that (F) , and therefore also the left-hand member of (E) , has the same sign as ( or ) when
Hence the following rule for finding maximum and minimum values of a function . ”

“ A binary form which has a square factor has its discriminant equal to zero. This can be seen at once because the factor in question being once repeated in both differentials, the resultant of the latter must vanish.
Similarly, if a form in k variables be expressible as a quadratic function of k – 1, linear functions X1, X2, ... Xk – 1, the coefficients being any polynomials, it is clear that the k differentials have, in common, the system of roots derived from X1=X2=...=Xk – 1=0, and have in consequence a vanishing resultant. ”

“ He observed also that all questions concerning lengths of curves, volumes enclosed by surfaces, and centres of gravity, can be formulated as problems of quadratures, and can thus be solved either in finite terms or by means of infinite series. In the Quadratura (1676) the method of integration which is founded upon the inversion-theorem was carried out systematically. Among other results there given is the quadrature of curves expressed by equations of the form y = xn (a + bxm) p; this has passed into text-books under the title “integration of binomial differentials” (see § 49) . ”

“ We proceed to indeterminate problems of the second degree: limiting ourselves to the consideration of the formula y2 = a + bx + cx2, where x is to be found, so that y may be a rational quantity. The possibility of rendering the proposed formula a square depends altogether upon the coefficients a, b, c; and there are four cases of the problem, the solution of each of which is connected with some peculiarity in its nature.
Case 1. Let a be a square number; then, putting g2 for a, we have y2 = g2 + bx + cx2. Suppose √ (g2 + bx + cx2) = g + mx ”

“ This integration gives us the last of the arbitrary constants which are functions of the coördinates and momenta without the time. The final integration, which introduces the remaining constant () , is also a quadrature, since the equation to be integrated may be expressed in the form
Now, apart from any such considerations as have been adduced, if we limit ourselves to the changes which take place in time, we have identically
and and are given in terms of by the differential equations of motion. When we have obtained integral equations, we may regard and as known functions of and . ”