Quadratic Equations. 5.^ 



for X it satisfies the equation. Also, both 2 and 3 are roots of the 

 equation x^—^x-\-io=^, for either of these values when substitu- 

 ted for X will satisfy the equation. 



The student must carefully note that this is an entirely different 

 use of the word rvot from that occuring in the expres.sions square 

 root, cube roof. etc. 



5, Equations of the second degree are often divided into the 

 two classes of complete and t^icomplete quadratics. A complete 

 quadratic is one which contains both the first and second powers of 

 the unknown, as x^-\-px=^q. An incomplete quadratic has the 

 first power of the unknown quantity lacking, and hence can al- 

 ways be placed in the fonii x"=q, where q is any algebraic quan- 

 tity conceivable. By some the adjectives affected and pure are 

 used in place of the words complete and incomplete respectively. 



6. Problem. To Solvk any Incomplkte Quadratic. 

 First, reduce to the form 



x''=q 

 by putting all the known quantities on the right hand side of the 

 equation and all the terms containing x" on the left hand side, 

 then dividing through by the coefficient of x\ 



Then take the square root of both sides of the equation, remem- 

 bering that every quantity has two square roots, and we obtain 



x=±^q 

 and the equation is solved. 



It might be thought that in taking the square root of both sides 

 of x'^q we should write 



But, by taking the signs in all possible ways, this give»i 



-\-x= + s/q 

 —x= — ^q 

 Jrx^-s/q 

 —x=-\-^q. 



