72 AlvGEBRA. 



16. "To Find the Condition that a Quadratic Equation 

 MAY hav:e: one Positive and one Negative Root. 



Since ^b=f 



if the roots are of opposite signs the absolute term must be negative. 



Since —(a-[-b) = e 



if the positive root is numerically the greater, e is negative and in 

 case the negative root is numerically the greater, e will be pcsitive. 



The conditio7i that a quadratic have roots of opposite signs is merely 

 that the absolue term be negative, but if the coefficient of x is nega- 

 tive the positive root is numerically the greater and if the coeffi- 

 cient of X is positive the negative root is numerically the greater. 



17. Examples. Discriminate the roots of the following equa- 

 tions; that is, tell by inspection whether the roots are real or im- 

 aginary, and if real, tell whether they are positive or negative. 



I. x^-|-8jr— 9=o. 



Jt:^ + 70Jt: + 1 200= o. 

 :r^— 4-r+4=o. 

 x^-f-iO-r+45=o. 

 jt:^— 8jr+20=o. 



X^=^\OX—2^. 

 JT^— I2X= — 27. 



2-^ 



18. In a manner similar to that of Arts. 14 — 16 the student 

 may determine the following : 



1. Find the condition that a quadratic equation may have 

 two roots numerically equal but of opposite signs. 



2. Find the condition that a quadratic equation may have 

 two roots which are reciprocals of each other. 



J. Find the condition that a quadratic equation may have 

 one root equal to zero. 



19. M1SCE1.1.ANEOUS Exercises in the Theory of Quad- 

 ratics. 



I. If a and b are the roots of x'-\-ex-\-f-=Q, find the value 

 oi a^^rb"" in terms of e and/. 



