| XXX | (94) | XXX |
BRA. L 94 volves at the fame time the fquare of that quantity, and whence the two values of y mull be imaginary or impoflithe produd of it multiplied by fome known quantity, ble, becaufe the root of — iif— cannot poffibly be afthen you have what is called a Quadratic equation; 4 figned. which may be refolved by the following Suppofe that 7the quadratic equation propofed to be R y l e. refolved is y —ay—b . x. “ Tranfport all the terms that involve the unknown then /"—ay+ —4 4 “ quantity to one fide, and the known terms to the o“ ther fide of the equatiom 2. “If the fquare of the unknown quantity is multi“ plied by any coefficient, you are to divide all the terms “by that coeffient, that the coefficient of the fquare of' and y— 2 If the fquare root of “ the unknown quantity may be unit. 4 3. “ Add to both fides the fquare of half the coeffici- b-- —1 cannot be extra#ed exa#ly, mull, in order “ ent prefixed to the unknown quantity itfe If, and the to determine the value of y, nearly you approximate to the “ fide of the equation that involves the unknown quan“ tity will then he a compleat fquare. 4. “ Extrafl the fquare root from both fides of the value of ^b-Ar—, by the rules in chap. 7. The fol“ equation; which you will find, oh one fide, always to lowing examples4 will illuftrate the rule for quadratic “ be the unknown quantity, with half the forefaid coef- equations. ’ “ ficient fubjoiiied to it; fo that, by tranfpofing this Examp. I. “ To find that number which if you “ half, you. may obtain the value'of the unknown quan- “ multiply by the produ# fhall be equal to the “ tity expreffed in known terms.” Thus, “ fquare of the fame number having 12 added to it.” Suppole jiz-j-aj=6 Call the number/, then Add the fquare ofto? J+^+^_ y7+i2—%y y71—8/=—12 both fides . . O 4 4 add the Sq.tranfp. of q,? —S/-)-i6=—I2-H6=4 Extra# the root, y+— extra# the R. /—4—_L:2 tranfp. /=4=±r2=6 or 2. Tranfpofe—, 2 42—• ■ Examp. II. “ To find a number fuch, that if you fubtra# it .from id, and multiply the remainder by The fquare root of any quantity, as + aa, may be ““ the number itfelf, the produft lhall give 21.’’ or -—a; and hence, “ All quadratic equations admit of2 two folutions.” In the laft example, after finding Call it/. Then , it may be inferred that/-)- that >> 10—/ X/~21 4 that is, IO/—yy—2 I / =+6+ — '4 or to —y i+ —4 fince - b+—y.~ tranfp. /**—ioy——;21 4 add thefq. of y,/1—10/4-25=—214-25=4 v ryf extra#, /—5=t=v/4=—2 b- T gives H-~—• as well as +4-^4-—5 x+ b-—-• 4 ^ 4 and /=5=t:2=7 or 3. There are therefore two values of y; the one gives/= Examp. III. “ A company dining together in an inn, IjJL —2 the other,/=— “ find their bill amounts to 175 ffiillings; two of them 4 4 2 were not allowed to pay, and the reft found that their Since the fquares of all quantities are pofitive, it ““ Ihares amounted 10 r. a man more than if all had is plain that “ the fquare root of a negative quantity is “ paid. Qu. Howtomany ?” imaginary, and cannot be affigned.” Therefore there are Suppofe their number x;werethenin ifcompany all had paid, each fome quadratic equations that cannot have any folution. man’s ftiare would have been ——: but now the lhare Eor example, fuppcfe x 1 1 of each perfon is ———, feeing x—2 is the number of then / —ay——3 a thofe that pay. It is therefore, by the queftion, -LTj— X 2 x and i75x-^i75x4-350 = iox*—20X that is, iox12—20x=350 extra# the root, v 2 =h and x1—2x=35 add 1 .^x —2x4-5=354-1=36 and y— -^-=±3^ i a extr. y" .. x—1—=1=6 x=i:±:6=7> or, —5,
