ax2-bx+c=a (x —m) (x — n), it follows that ax2+ bx+c=a (x+m) (x+n). I. We know that when b2 is greater than 4ac, III. When b2 is less than 4ac, ax2+ bx+c cannot be divided into factors. Therefore y is 0 when 2ax — b\2=k2, or when 2ax—b= ±k: Now, because b2 is less than b2+4ac, b is less than b2+4ac, therefore n is a negative quantity. Leaving, for the present, the consideration of the negative quantity, we may decompose (3) into factors by means of the general formula p2 -q2 = p q p+q, which gives The following are some examples, of the truth of which the student] should satisfy himself, both by reference to the one just established, and by actual multiplication: If we collect together the different results at which we have arrived, to which species of tabulation the student should take care to accustom himself, we have the following: These four cases may be contained in one, if we apply those rules for the change of signs which we have already established. For example, the first side of (C) is made from that of (A), by changing the sign of c; the second side of (C) is made from that of (A) in the same way. We have also seen the necessity of taking into account the negative quantities which satisfy an equation, as well as the positive ones; if we take these into account, each of the four forms of ax2+bx+c can be made 2a In the cases marked (B) (C) and (D), the results are and in all the four cases the form of a x2 + bx + c which is used, is the same as the corresponding form of a (x−m) (x − n) and the following results may be easily obtained. In (A'), both m and n, if they exist at all, are negative. I say, if they exist at all, because it has been shown that if b2-4ac is negative, the quantity ax + bx + c cannot be divided into factors at all, since √b2-4ac is then no algebraical quantity, either positive or negative. In (B'), both, if they exist at all, are positive. In (C') there are always real values for m and n, since b2 + 4 ac is always positive; one of these values is positive, and the other negative, and the negative one is numerically the greatest. In (D') there are also real values of m and n, one positive, and the other negative, of which the positive one is numerically the greatest. Before proceeding any further, we must notice an extension of a phrase which is usually adopted. The words greater and less, as applied to numbers, offer no difficulty, and from them we deduce, that if a be greater than b, a c is greater than b-c, as long as these subtractions are possible, that is, as long as c can be taken both from a and b. This is the only case which was considered when the rule was made, but in extending the 12, or meaning of the word subtraction, and using the symbol 3 to stand for 5-8, the principle that if a be greater than b, α- c is greater than b c, leads to the following results. Since 6 is greater than 4,6 12 is greater than 4 -6 is greater than -8; again 6-6 is greater than 4-6, or 0 is greater than 2. These results, particularly the last, are absurd, as has been noticed, if we continue to mean by the terms greater and less, nothing more than is usually meant by them in arithmetic; but in extending the meaning of one term, we must extend the meaning of all which are connected with it, and we are obliged to apply the terms greater and less in the following way. Of two algebraical quantities with the same or different signs, that one is the greater which, when both are connected with a number numerically greater than either of them, gives the greater result. tive when m and x- n have the same, and negative when they have different signs. This last can never and n, that is, when x is algebraically happen except when x lies between m greater than the one, and less than the other. The following table will exhibit this, where different products are taken with various signs of m and n, and three values are given to x one after the other, the first of which is less than both m and n, the second between both, and the third greater than both. Thus 8, be- n = 6 is said to be greater than cause 20 6 is greater than 20 is greater than -4, because 60 is greater than 6 4; 12 is greater 30, because 40+12 is greater than 40 30. Nevertheless 30 is said to be numerically greater than +12, because the number contained in the first is greater than that in the second. For this reason it was said, that in (C'), the negative quantity was numerically greater than the positive, because any positive quantity is in algebra called greater than any negative one, even though the number contained in the first should be less than that in the second. In the same way 14 is said to lie between 3 and 20, being less than the first and greater than the second. The advantage of these extensions is the same as that of others; the disadvantage attached to them, which it is not fair to disguise, is that, if used without proper caution, they lead the student into erroneous notions, which elementary works, far from destroying, confirm, and even render necessary, by adopting these very notions as definitions; as for example, when they say that a negative quantity is one which is less than nothing; as if there could be such a thing, the usual meaning of the word less being considered, and as if the student had an idea of a quantity less than nothing already in his mind, to which it was only necessary to give a name. some The product - m x n is posi +3 m = n = The student will see the reason of this, and perform a useful exercise in making two or three tables of this description for himself. The result is that x-m х In is negative when a lies between m and n, is nothing when x is either equal to m or to n, and positive when x is greater than both, or less than both. Consequently, a (x − m) (x n) has the same sign as a when x is greater than both m and n, or less than both, and a different sign from a when x lies between both. But whatever may be the signs of a, b, and c, if there are two quantities m and n, which make a x2 + bx + c = a (x − m) (x — n), that is, if the equation ax2+ bx+c=0 has real roots, the expression ax2+ bx+c always has the same sign as a for all values of x, except when lies between these roots. It only remains to consider those cases in which ax2 + bx + c cannot be decomposed into different factors, which happens whenever b2-4ac is 0, or negative. In the first case when b2-4ac = 0, we have = ax2 + bx + c = a ( x + 2 ) as! - bx + c = a a. = and as these expressions are composed possible, except when c and a have different signs. In this case, that is, when the expression assumes the form ax2-c, it is the same as a ( x − √(x + √). The same result might be deduced by = When a = 0, the expression is reduced to bx + c, which is made equal to nothing by one value of x only, that с is-음. If we take the general expressions for m and n, and make a = 0 in ·b + √√b2 - 4ac them, that is, in solves the equation, still that the greater the number which is taken for x, the tained. The use of these expressions more nearly is a second solution obis to point out the cases in which there is anything remarkable in the general problem; to the problem itself we must resort for further explanation. The importance of the investigations connected with the expression ax2 + bx + c, can hardly be over-rated, at least to those students who pursue mathematics to any extent. In the higher branches, great familiarity with these results is indispensable. The student is until he has completely mastered the detherefore recommended not to proceed tails here given, which have been hitherto too much neglected in English works on algebra. с In solving equations of the second degree, we have obtained a new species of cannot be solved at all. We refer to those result, which indicates that the problem results which contain the square root of a negative quantity. We find that by multiplication the squares of c-d and 2cd+d. Now either c of d c are the same, both being c2 d or d is positive, and since they both have the same square, it appears that the squares of all quantities, whether positive or negative, are positive. It It is therefore and absurd to suppose that there is any quantity which a can represent, and which satisfies the equation 2 we find as the results a2, since that would be supposing that x2, a positive quantity, is equal to the negative quantity - a. The solution is then to show an instance in which such a resaid to be impossible, and it will be easy sult is obtained, and also to show that it arises from the absurdity of the problem. 2α These have been already explained. The first may either indicate that any value of x will solve the problem which produced the equation ax2 + bx + c = 0, or that we have applied a rule to a case which was not contemplated in its formation, and have thereby created a factor in the numerator and denominator of x,which, in attempting to apply the rule, becomes equal to nothing. The student is referred to the problem of the two couriers, solved in = Let a number a be divided into any the half, and the other less. Call the two parts, one of which is greater than parts must be a. Multiply these parts to- part of algebra is established which is of great utility. It depends upon the α gether, which gives (¦ + x ) (1⁄2-), or -X (2)2 – æo. As a diminishes, this product increases, and is greatest of all when x 0, that is, when the two parts, = α into which a is divided, are and, or Hand In this a 2 2' or and a number a can never be 4 divided into two parts whose product is fact, which must be verified by experi ence, that the common ruies of algebra may be applied to these expressions without leading to any false results. An appeal to experience of this nature appears to be contrary to the first principles laid down at the beginning of this work. We cannot deny that it is so in reality, but it must be recollected that this is but a small and isolated part of an immense subject, to all other branches of which these principles apply in their fullest extent. There have not been wanting some to assert that these symbols may be used as rationally as any others, and that the results derived from them are as conclusive as any reasoning could make them. I leave the student to discuss this question as soon as he has acquired sufficient knowledge to understand the various arguments: at present, let him proceed with the subject as a part of the mechanism of algebra, on the assurance that by careful attention to the rules laid down, he can never be led to any incorrect result. The simple rule is, apply all those rules to such expressions as a a+ √-b,&c. which have been proved to hold good for such quantities as a a+ √√, &c. Such expressions as the first of these are called imaginary, to distinguish them from the second, which are called real; and it must always be recollected that b*, there is no quantity, either positive or negative, which an imaginary expression can represent. b It is usual to write such symbols b in a different form. To the equation − b = b × (−1) apply the rule derived from the equation √xy √y, which gives√ b = √b× √√ − 1, of which the first factor is real and the second imaginary. Let We have shown the symbol -α to be void of meaning, or rather = c; then self-contradictory and absurd. Never- this way all theless, by means of such symbols, a arranged that O b=c√1. In expressions may be so I shall be the only The general expressions for m and n give imaginary quantity which appears in 46 as the roots of 2 — ax + b −0. 2 them. Of this reduction the following are examples: E 2 |