ALGEBRA. 



Let x be the time required. 



In one day, 1 performs part of the 

 o 



work ; therefore, in x days, he performs 



^- parts of it ; and in the same time, B 

 8 



performs parts of it ; and calling the 

 work 1, 



s+Io^ 1 ' 



10j'+8x=80 

 18x=80 



80 .8 .4 



Prob. 3. Jt and B play at bowls, and Jl 

 bets B three shilling* to two upon every 

 game; after a certain number of games, 

 it appears that A has won three shillings ; 

 but hud he ventured to bet five shillings 

 to two, and lost one game more out of 

 ,;ne number, he would have lost 

 thirty shillings : how many games did 

 they play ? 



C be the number of games 

 Let x \ .1 won, 



y the number B won, 

 then 2 JT is what -7 won of B, 

 and 3 y what B won of . /. 



2 I- 3 y=3, by the ques- 

 tion ; 



would win on 

 ' "i tlie 2 d supposition 

 . 5, fl would win, 

 5 _(_ 5 2 .r + 2=30, by 

 the question ; -, 



or 5 y 2 r = 30 5 2=23, 

 therefore, 5 y 2 x=23 



and 2 .r 3 .y=3 

 by addition, 5 y 3 y=26 

 2y=26 

 y = 13 

 2.r =3 +3^=3+39 = 42 



_ 

 x _ i 2 



' 



.r -j- y = 34, the number of 



gamei 



ON qrADRATIC F.(U ATIO.VS 



\Vhcn the tenns of an equation involve 

 the square of an unknown quantity, but 

 the first power does not appear, the value 

 of the square is obtained by the ].: 

 ing rules ; and by extracting the square 

 root on both sides, the quantity itself is 

 found. 



. 1. Let 5 J-= 45^0 ; to find x. 

 By trans. 5 a" = 45 



~ 9 



therefore. r 



The signs + and are both prefixed 

 to the root, because tin <>t of a 



quantity may be either positive or nega- 

 Tlir sign of .1- may also be nega- 

 ilf .r will be eitli 



tive ; but stil 

 -}- 3 or 3 



Ex. 2. Let a x*=b cd ; to ! 

 bed 



If both the first and second powers of 

 the unknown quantity be found in an 

 equation : Arrange the terms according 

 to the dimensions of the unknown quanti- 

 ty, beginning with the highest, and trans- 

 pose the known quantities to the other 

 side ; then, if the square of the unknown 

 quantity be affected with a co-efficient, 

 divide all the terms by this co-efficient, 

 and if its sign be negative, change the 

 signs of all the terms, that the equation 

 may be reduced to this form, x 1 /> x= 

 q. Then add to both sides the square 

 of half the co-efficient of the first power 

 of the unknown quantity, by which means 

 the first side of the equation is made a 

 complete square, and the other consists 

 of known quantities; and by extracting 

 the square root on both sides, a simple 

 equation is obtained, from which the value 

 of the unknown quantity may be found. 



Ex. 1. Let a -f p x=q ; now, we 



/ 



know that x 1 -|- p a- -j- is the square 

 4 



of x -f'-, add therefore/!? to both sides, 

 2 4 



and we 



iL = gr -j. t ; 

 4 4 



then by extracting the square root on 

 both sides, 



.v + -= J (q +f J nd by trans. 



In the same manner, if j p r=q t x 

 is found to be^i J Iq + ^ \ 



Ex. 2. Let xa 12 x + 35=0; to find x. 

 My transposition, x* 12 a- = 

 adding the square of 6 to both sides of 

 the equation, 



x _ 12 x + 36 = 36 35 = 1 ; 

 then extracting the square root on both 

 sides, 



