ALGEBRA. 325 



simply by extracting the square root of both of its members. Thus, 

 from the equation 10 = 108 2 x 2 , we obtain 2x* = 108 10 ; or 

 2 x 2 = 98 ; or x 2 = 49 ; or x = 7. In this case we may have 

 x = 7, or x = 7 ; since a negative quantity multiplied by itself 

 produces a positive square. As the square root of any quantity is 

 denoted by the radical sign, (>/ ), we might have written above, 



The square root of a monomial, is also a monomial : but if we 

 multiply x + a by x + a, we shall have (x + a) 3 = x 2 + 2 ax + a 3 ; 

 that is, the square of a binomial, is made up of the square of the first 

 term, plus twice the product of the two terms, plus the square of the 

 last term. This suggests the rule for extracting the square root of a 

 polynomial ; which we have no room here to present. Hence, to 

 resolve a complex quadratic equation, when reduced to the regular 

 form, a? 3 + ax = b", we must consider ax as twice the product of the 

 two terms of a binomial root ; and x being one of them, a will ne- 

 cessarily be the other. We must therefore add the square of 5 a to 

 each member of the equation ; making x 2 + ax + I a 2 = b + $ a 2 ; 

 and the first member will then become a trinomial and perfect square; 

 while the second member will contain only known quantities. Then, 

 extracting the square root of each member, we have x + 5 a = 

 */b + | a a ; from which, as a simple equation, the value of x may 

 readily be found. For example, if we have x*+ 6 a? = 27, then is 

 #2_j_ 6a? + 9 = 27 + 9 = 36; and x + 3 = 6, or x = 6 3 

 == 3, or 9. 



4. The theory of Powers and Roots in general, comes next in 

 order, as a preparation for the more general study of equations. If 

 we form the successive powers of the binomial a + b, we shall have 



[a 



== a 3 + 3 a* b + 3 a b 2 + b 3 . 



a * 4. 4 a 3 b + 6 a 8 6 3 + 4 a b 3 + b*. 

 In the formation of these powers, we observe certain remarkable 

 laws, which have been generalized by Newton, in the binomial the- 

 orem. We see that the number of terms in the power, is one greater 

 than its exponent. The exponents of the leading factor, a, go on 

 diminishing by unity from term to term ; while those of the succeed- 

 ing factor, 6, go on increasing, according to the same law. And to 

 form the coefficient of any term, we multiply the co-efficient of the 

 preceding term by the first exponent in that term, and divide the 

 product by the number denoting the place of that term, counting 

 from the first. 



By these same rules, we may develope the powers of any other 

 binomial. Thus, to develope (2 x + y) 3 , we write 2 x instead of a, 

 and y instead of b ; and the result becomes, (2 x) 3 + 3 (2 x) 2 y + 3 

 (2 x) y* + y 3 ; or by reduction we have (2 x + y} 3 S x 3 + 12 x*y 

 _{_ Q x y* + y 3 . Roots, in general, are denoted by the radical sign, 

 (x/~~~) with the index of the root written above and on the left ; ex- 

 cept the square root, whose index is understood, but not written. As 

 we multiply the exponent, in raising to a power, so we may divide 

 the exponent, to extract the root ; thus forming a fractional exponent. 



2E 



