of the third form, whofe roots become impoffible. 2. It is obvious that, in the two first forms, one of the roots must be positive, and the other negative. 3. In the third form, if a2, or the square 4 of half of the coefficient of the unknown quantity be greater than b2, the known quantity, the two roots will be positive. a2 If be equal to 62, the two roots then be 4 come equal. But if, in this third cafe, a2 is less than 4 62, the quantity under the radical sign becomes negative, and the two roots are therefore impoffible. This may be easily shewn to arife from an impoffible fuppofition in the original equation. 4. If the equation, however, exprefs the relation of magnitudes abstractly confidered, where a contrariety cannot be fuppofed to take place; the negative roots cannot be of use, or rather there are no fuch roots; for then a negative quantity by itself is unintelligible, intelligible, and therefore the fquare root of a pofitive quantity must be positive only. Hence, in the two first cafes, there will be only one root; but, in the third, there will be two. For, in this third cafe, x2-ax= -b2, or ax-x2-b2, it is obvious that may be either greater or less than ļa, and yet a-x may be pofitive; and hence a-xxx=ax-x2 may alfo be positive, and may be equal to a given pofitive quantity b2; therefore the fquare root of x2—ax+ 4a2 may be either x-a, or a-x, and both these quantities alfo pofitive. are the fame two positive roots as were ob tained by the general rule. The general rule is ufually employed, even in questions where negative numbers cannot take place, and then the negative roots of the two firft forms are neglected. Sometimes Sometimes even, only one of the positive roots of the third cafe can be used, and the other may be excluded by a particular condition in the question. When an impoffible root arises in the folution of a question, which has been refolved in general terms, the neceffary limitation of the data will be difcovered. When a question can be fo ftated, as to produce a pure equation, it is generally to be preferred to an adfected. Thus the queftion in the preceding section, by the most obvious notation, would produce an adfected equation. II. Solution of Questions producing Quadratic Equations. The expreffion of the conditions of the queftion by equations, or the stating of it, and the reduction likewise of these equations, till we arrive at a quadratic equation, involving only one unknown quantity and its fquare, are effected by the fame rules which were given for the solution of fimple equations, in Chap. III. EX EXAMPLE II. One lays out a certain fum of money in goods, Therefore by question, And by mult. and tr. 5y+100y=2400 6 y2+100y+502=2400+2500=4900. 7y+50=±√4900=70 8y=70-50=20, or -120. The answer is 20l. which fucceeds. The other root, 120, has no place in this example, a negative number being here unintelligible. Any quadratic equation may be refolved also by the general canons at the beginning of this fection. That arifing from this queftion, (No. 5.), belongs to Cafe 1. and a=100, b2=2400; therefore, y 100 4 +2400=20, or -120, as before: EXAMPLE III. What two numbers are thofe, whofe difference is 15, and half of whofe product is equal to the cube of the leffer? Divide by x and mult. by 2.4x+15=2x2 The numbers, therefore, are 3 and 18, which answer the conditions. This is an example of Cafe 2d, and the negative root is neglected. A |