Theory of Quadratics. 



69 



j:^—2ax-\-a''=o, where it is seen that —2a is the sum of two roots 

 with the opposite sign, and a^ equals their product. 



II. Examples. We can now form a quadratic equation which 

 shall have any two roots we desire. Suppose we wish a quadratic 

 whose roots shall be 3 and 5. Then <?=— (^34-5;=— g, and 

 /=3 X 5= 15- Then the equation is 



1. Form the equation whose roots shall be 3 and —5. 



2. " " " " '' " —3 and 5. 

 J. " " " " " " -3 and -5. 



4- '' '' " *' " " a and - . 



a 



5. " " ** " " " 2-f v/3and2 — v/3 



6. " *' " " " " Vyand— Vy- 



7. " *' " " " " —5 and o. 



8. '' '' " " " " 6 and 6. 



9. " " " " " " o and o. 



12. The student should not understand that there is only one 

 method of solving the quadratic equation. The fact is that the 

 result may be reached in a great variety of ways, that of IV, Art. 

 9, merely being one among a great number. But many of the 

 diiferent methods that have been proposed are, in the last analysis, 

 essentially the same, and they all resolve themselves into the one 

 principle of reducing the quadratic to some form of a simple equa- 

 tion. We give a few methods of solution to show the student 

 what a variety of means may be made use of in such work. 



(a). By reduction to a7i incomplete quadratic. 



x^-\-ex-\-f=-o. 

 Suppose x=y—^e, where jk is a new unknown quantity ; then 

 the equation becomes 



0'— i^/ + ^(y—\^) +/=o, 



or y^-ey+y-^-ey-^e'+f^o, 



or _^/2_i-^+/"=o. 



