34 



A new Quadratic Theorem. 



[Jan. 



Extract the square root from the following pentanomial, and 

 prove it by involution : 



x* - 4 X s + 20 x ■• - 32 r + 64 



Should the operator wish to save himself the tedious multipli- 

 cations consequent to this proof by involution, he may either 

 substitute a letter according to the binomial theory of Newton, 

 or he may use the following more simple and concise method : 



Multiply the first by the last term of the square root, and 

 double the product : secondly, square the mediate term of the 

 root, and subtract from the mediate terms of the square quan- 

 tity ; if nothing remain, we may be certain the calculation is 

 correct. Thus in this last example we multiply the first term of 

 the root -f *'• by the last term + 8=4- 8.r°-, which doubled 

 = + 16 x*, and the mediate term — 2 x squared = + 4 x*, 

 which added = + 20 x« ; subtracted, leave nothing. The ad- 

 vantage of this method may be well illustrated by extracting the 

 square root according to the rule at present recommended by 

 authors, and then involving the root back again to the square 

 quantity, when it will be found that, according to the theorem 

 herein recommended, the evolution and involution may be per- 

 formed in two lines, instead of 12, required by the other. This, 

 independent of its simplicity and accuracy,' must, 1 presume, 

 recommend its adoption. 



Extract the square root from 



a* + 4 a x I + 4 a b + 4 x* } + 8 b x + 4 b* 

 a + 2 x 1+26+0 



Here to measure the mediate terms + a x + 2 b = 2 a b 

 and + 2 a b doubled = 4 a b, and + 2 x squared == 4 r*, which 

 subtracted from the mediate terms of the compound quantity 

 leaves nothing. The square root of a pentanomial or a hexano- 

 mial never exceeds a trinomial. In a pentanomial the root is 

 extracted from the first, second, and fourth terms : in a hexano- 

 mial, from the first, second, and fifth terms. Indeed a penta- 

 nomial is nothing more than an abbreviated hexanomial. 



Extract the square root of the following hexanomial : 

 « 3 + 4 a b I - 2 a r + 4 6* j - 4 b x + x* 

 a + 2 b \ j _ . x +o square root. 



