CHAPTER V. 



THEORY OF QUADRATIC EQUATIONS AND QUADRATIC FUNCTIONS. 



I. It follows immediately from the definition (I Art. 4) that 

 every rational integral quadradic function of x is of the form 



where f, 11 and r stand for any algebraic numbers whatever, posi- 

 tive or negative, integral or fractional, commensurable or 

 incommensurable. 



If we take the typical quadratic equation 



ax''-\-bx=^c 

 and transpose the c to the left-hand side of the equation it becomes 



ax~-\-bx — r=o. 

 This can obviously be said to be of the form 



lx'-\-7ix-\-r^=o 

 and consequently a quadratic equation may be defined as an equa- 

 tion which can be placed in the form of a rational integral quad- 

 ratic function equal to zero. 



Since a root of an equation has been defined as any expression 

 which substituted for the unknown will satisfy the equation, there- 

 fore it is evident from the form 



ax^-\-bx — <:=o 

 that a root of a quadratic equation may also be stated to be an 

 expression which substituted for x causes ax--\-bx—c to equal 

 zero; that is, causes the function* oi x to vanish. 



Hence we may say : A quadratic equation is a7iy equation which ^^ — ^ 

 can be put in the form of a 7'ational iyitegral quadratic functio7i equal \tf\< 

 to ze7'o, a7id a root of it is a7iy expression ivhich, substituted for x, 

 causes the fu7ictio7i of x to vanish. 



Thus the equation .r^— 3.^=10, whose roots are 5 and —2, when 

 placed in the form of a function of x equal to zero, becomes 



x^ — 3j; — 10=0. 

 It is now seen that the roots are such quantities that, when sub- 



*Becaus8 of the ai-ray of adjectives in the expression ' rational integral <inadrati( 

 function of x " we shall often, for the remainder of this chapter, use the expression "func- 

 tion of X " in its place. 



