52 Algebra. 



Now rationalizing with respect to .i , by squaring both sides of 

 the equation, it becomes 



Finally transposing and collecting terms, we have 



which is a quadratic equation. While the equation (5, a) has 

 been reduced to the quadratic form (5, d) by apparently legitimate 

 processes, yet we will find that the integralization and rationaliza- 

 tion- of an equation with reference to the unknow'n quan- 

 tity has in general an eitect on the solution of the equation 

 ivhich it is necessary to take into account, and which renders it pos- 

 sible that the values of x which satisfy (5, b) may not be identical 

 with those that satisf}^ (5, a). For this reason the treatment of 

 those equations which require the operation of integralization or 

 of rationalization before they are in the quadratic form, is reserved 

 for Chapter VI. 



3. Typical Forms ok the Quadratic. It is evident that 

 equations (\), (2), (^3^, (/\.) and (^5, b), or any other quadratic 

 equations which can be imagined, may all be said to be of the 

 typical form, 



ax^-\-bx=^c, (6) 



where a, b and c are supposed to stand for any numbers whatever, 

 either integral or fractional, positive or negative, or commensur- 

 able or incommensurable. Hence ax^'-ybx—c is said to be a 

 typical form of the quadratic equation. 



If we suppose the quadratic equation to be divided through by 

 the coefficient of x'^ the result will be of the form 



x'-\-px=q, (7) 



where p and (j are supposed to be any algebraic quantities what- 

 ever, fractional or integral, positive or negative, commensurable 

 or incommensurable. This is the second typical form of the 

 quadratic equation, and one which is much used. 



4. Definition. A Root oi^n equation is any value of the un- 

 known quantity which satisfies the equation. 



Thus \ isa root of the equation 3;r— 6=0, for when substituted 



