10 ANALYTIC GBOMETi;) [Cn. I. 



The nature of tin- root* (5) lrp.-nU upon the 

 uiuler the radical sign, i.e., upon ft* 4 ac, giving three 

 CMOS to be considered, v 



if P 4 ac > 0, then the roots are both real and unequal, 



if 6* 4 ac = 0, then the roots arc lth real and equal, (6) 



if ft 1 4 ac< 0, thru tin* roots are both imaginary. 



Thus the character of the roots of a given quadratic equa- 

 may be drtrrmim ! wit hout actually solving the equation. 

 la merely calculating the value of tin expression b* 4 ac. 

 Tiiis important expression is called the discriminant of the 

 quadratic equation ; when equated to zero it states the con- 

 dition that must hold among the coefficients if the equation 

 has equal roots. 



EXERCISES 



1. Show which of the following equalities are identities : 



(1) r * _ 4 x + 4 = 0; (4) (p + ?) = p> + 7 + 3pq(p + 7 ); 



(2) ( + /)( - = - **; (">) ** + *x + 6 = (x + 3)(* + 2). 



a '^ = a-a0 + ; 

 a + p 



2. I Determine, without solving the equation, the nature of the roots of 



3 X + 8z + 1 =0. 



I.UTION. Since 6* - 4 ac = 64 - 12 = 52, i.., is positive, therefore 

 the roots are real and unequal ; again, since a, 6, and c are all positive, 

 therefore both roots are negative (cf. eq. (4), Art. 0). 



3. Without solving the equation, determine the character of the 

 rootaof 8**-3z+ 1 =0. 



4. Given the equation z 1 - 8x - m(z + 2z* + 4) = 5 a: 2 + 3. 

 1 the roots. For what values of m are these roots equal? 



5 Determine, without solving, the character of the roots of the 

 equations : 



(1) 5** -2* -1-5 = 0; (2) z* 4 7 = 0; (3) 3f*-f=19. 



