ALGEBRA. 



It appears from the second example, 



x 1 

 that a trinomial a 1 a x + ,in which 



four times the product of the first and 

 last terms is equal to the square of the 

 middle term, and a complete square, or 



X 2 



aZ x ~4~ X4=a*o-2. 



The method of extracting the cube 

 rqot is discovered in the same manner. 



a3+3 a- 



(a+b 



3 a 1 b+ 3 a ft s + fr 

 3 a 2 6.4.3 a b--\-bi 



3 a*) 



The cube root of a3+3 a 1 6+3 

 is a+6; and to obtain rt+6 from this 

 compound quantity, arrange the terms as 

 before, and the cube root of the first term, 

 fl?, is a, the first factor in the root : sub- 

 tract its cube from the whole quantity, and 

 divide the first term of the remainder by 3 

 a 2 , the result is b, the second factor in the 

 root: then subtract 3 a2 6+3 a 62+63 

 from the remainder, and the whole cube 

 of a-\-b has been subtracted. If any 

 quantity be left, proceed with a+b as a 

 new a, and divide the last remainder by 

 3 . a+6) 2 for a third factor in the root ; 

 and thus any number of factors may be 

 obtained. 



OX SIMPLE EQUATIONS. 



If one quantity be equal to another, or 

 to nothing, and this equality be expressed 

 algebraically, it constitutes an equation. 

 Thus, x a =b x is an equation, of 

 which x a forms one side, and b x 

 the other. 



When an equation is cleared of frac- 

 tions and surds, if it contain the first pow- 

 er only of an unknown quantity, it is call- 

 ed a simple equation, or an equation of one 



dimension : if the square of the unknown 

 quantity be in any term, it is called a 

 quadratic, or an equation of two dimen- 

 sions ; and in general, if the index of the 

 highest power of the unknown quantity 

 be n, it is called an equation of n dimen- 



In any equation quantities may be trans- 

 posed from one side to the other, if their 

 Mg-718 be changed, and the tioo sides toill still 

 be equal. 



Let x + 10=15, then by subtracting 10 

 from each side, x + 10 10 = 15 10 

 or x = 15 - 10. 



Let x 4=6, by adding 4 to each side, 

 x 4 + 4=6 + 4, or o-=6+3. 



If x a + b=y ; adding a b to each 

 side, x a + 6 + a b = y -\- a b ; 

 or a: = y + a b. 



Hence, if the signs of all the terms on 

 each side be changed, the two sides will 

 still be equal. 



Let x =6 2 x ; by transposition, 

 07= x-\-a ; or a 07=2 x b. 



If every term, on each side, be multiplied 

 by the same quantity, the results -will be equal. 



An equation may be cleared of frac- 

 tions, by multiplying every term, succes- 

 sively, by the denominators of those frac- 

 tions, excepting those terms in which the 

 denominators are found. 

 5 x 



Let 3 07 + -7-= 34; multiplying by 4, 



12o-+5o; = 136, or 17 07 = 136. 



If each side of an equation be divided by 

 the same quantity, the results will be equal. 



Let 17 x 136 ; then 07 = -^=- = 8. 



If each side of an equation be raised t* 

 the same power, the results ivill be equal. 



Let 07* = 9 ; then 07 = 9 X 9 = 81. 

 Also, if the same root be extracted on 

 both sides, the results will be equal. 



Let x = 81 ; then o^ = 9. 



To find the value of an unknown quantity 

 in a simple equation. 



Letthe equation first be cleared of frac- 

 tions, then transpose all the terms which 

 involve the unknown quantity to one side 

 of the equation, and the known quantities 

 to the other, divide both sides by the co- 

 efficient, or sum of the co-efficients, of 



