Theory of Quadratics. 71 



13. Discrimination of the Roots of the Quadratic Equa- 

 tion. The roots of the equation 



are a=— |<f+ Vi^^— /and x^—^e—\/\e'—/. 



(a). If i^— / is positive there are two real and unequal roots. 



(d). If ^e'—fis negative there are two imaginary roots. 



(cj. If i^— / is zero the two values of x each reduce to —^e 

 and the two values of x are real and equal. 



(d). li \r—f is a perfect square the two roots are rational, if 

 e is rational. 



(e). If \e-—fis not a perfect square the roots are irrational. 



The expression \e^—f is called the Discriminant. 



The case where \e^—f is zero deserves further attention. If 

 J^— y=o then \e^=f and the equation x- + ^ji;H-y=o becomes 



x'-\-ex-\-\e'=o 

 or (x+\e)(x-\-\e)=o. 



Whence we see that when a quadratic equation has two equal roots 

 the function of x is a complete square. 



14. To Find the Conditions that a Quadratic Equation 

 MAY HAVE TWO POSITIVE RooTS. Represent the roots by a and b. 



Then since —(a-{-b)^e 



if the roots are both positive the coefficient of x must be negative. 

 Also since ^b=f 



if the roots are both positive the absolute term must be positive. 



Hence the full condition that both the roots of a quadratic be posi- 

 tive is that the coefficient of x be negative and the absolute term 

 positive. 



15. To Find the Condition that a Quadratic Equation 

 MAY HAVE Two Negative Roots. Represent the roots as before. 



Then since — (a-^b)^=e 



if both roots are negative the coefficient of x must be positive. 

 And since ab=f 



if both roots are negative, the absolute term must be positive. 



Hence the full coyidition that both the roots^ o£ a quadratic be neg- 

 ative is that the coefficie?it of x^be positive^nd the absohdejenjp me^ 



