ALGEBRA. 



Let x be the time required. 

 In one day, JI performs - part of the 

 work ; therefore, in x days he performs 

 |j parts of it ; and in the same time, B 



performs -rr parts of it ; and calling the 

 work 1, 



10 x+8x=80 

 18 or=80 



80 . 8 .4 



Prob. 3. Ji and B play at bowls, and JI 

 bets B three shillings to two upon every 

 game; after a certain number of games, 

 it appears that Ji has won three shillings ; 

 but had he ventured to bet five shillings 

 to two, and lost one game more out of 

 the same number, he would have lost 

 thirty shillings : how many games did 

 they play ? 



C be the number of games 

 ***{ .4 won, 



y the number B won, 

 then 2 x is what JI won of B, 

 and 3 y what B won of A. 



%x' 3 #=3, by the ques- 



tion ; 



_ 9 C JI would win on 

 x A * t the 2d supposition 

 y -\- 1 . 5 y li would win, 

 5y + 5 2x + 2=30, by 

 the question ; 



or 5 y _ 2 x=30 5 2=23, 

 therefore, 5 y 2 x=23 



and 2 x 3 y=3 

 by addition, 5 # 3 #=26 

 2 y = 26 

 yLj3 



2#= 3 + 3#=3-|-39 = 42 

 a: = 21 



x -j- y = 34, the number of 

 games. 



OJf QUADRATIC EQUATIONS. 



When the terms 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 preced- 

 ing rules ; and by extracting the square 

 root on both sides, the quantity itself is 

 found. 



Ex. 1. Let 4 x 2 45=0; to find*. 

 By trans. 5 x 2 .= 45 

 x 2 =9 

 therefore, x = 



The signs + and are both prefixed 

 to the root, because the square root of a 

 quantity may be either positive or nega- 

 tive. The sign of x may also be nega- 

 tive ; but still x will be either equal to 

 + 3 or 3, 



Ex. 2. Let a x 2 =& c d; to find a-. 



\ 

 a ) 



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-effici : nt, 

 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* p x =0: 

 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 x 1 + p x = q ; now, we 

 know that x 2 + p x + is the square 



ofx+, add therefore, to both sides, 

 and we have x 1 + p x -\.*L = q +P1 . 



then by extracting the square root on 

 both sides, 



trans. 



In the same manner, if x 1 p 

 is found to be 



Ex. 2. Let a: 1 12 x -f 35=0 ; to find x. 

 By transposition, x 2 12 x = 35, and 

 adding the square of 6 to both sides of 

 the equation, 



x 1 - 12 x + 36 =r 36 35 = 1 ; 

 then extracting the square root on both 

 sides, 



