41 B Mr. hell 1 ns’s M-.thod cfi finding the 
2a + b=p, aa+iab = q, and aab~r\ which values being 
written in our theorem, we have se ( = llzEE. ) = 
•}jiaa -f- Aaah + m b 4 'icbb— qaah 2<r aa — Anah -}- 1alf> 
< 6 ab -bzjjb— baa — I 2 ab a«a — 4^-j- ibb 
-a. Q.E.D, 
E X A M P L E I.. 
It the equation + 5 X * ~ 3 2x + 36 = o has two equal roots,, 
it is propoied to find them by the above theorem. 
Here p = -5, q=> -32, and r= —36; thefe values being 
written in the theorem, we have x ~ 3 6 __ 160 + 3 2 4 
• •• • 2x25 -Ox -32 50+192 
= Jjf = 3 * which being written for x, the equation becomes 
S + 20 - 64 + 36, which is evidently = o ; confequently 2 and 2 
are roots of it. 
Other wife, 2, the value of x given by the theorem, being 
written for it in the quadratic equation gx 2 -f io,v — 32 = o, the 
refult is 12 + 20-32 = 0. 
Or, ^dividing the given cubic by the quadratic we 
have * 2 - 4* + 4). a; 3 + 5V 2 - 3.2* + 36 (x + 9 ; therefore the three 
roots are 2, 2, and - 9. 
Example il. 
Given v 3 _i_ 4®oo . 
+ T ~ 9261 = ° 5 311 equation which has equal roots, 
to find them. 
Here q — o, and the theorem gives ZJ. 6o ?° x f 9 = ~ 2 Q which 
... . b 200x9261 Ti» wnicl1 
value being written for * the equation vanilhes. 
THEOREM 
