68 Algebra. 



10. Theorem. Whe7i a quadratic eqiiatioji is iri the form 

 x^ -\- ex -\-/= o , the coeJJicie7it of x ivith its sign changed equals the 

 sum of the tivo roots,. and the absolute term equals the product of the 

 tivo roots. 



The two roots of the equation 



x''-\-ex-{-f—o 

 are —\e-\-s/\e^—f 



and —\e—sj\(f—f 



— e-\- o 

 Adding them, their sum is seen to be —e, or the coefficient of x 

 with its sign changed. 



Multiplying the two roots together, recognizing the product of 

 a sum and difference, we obtain 



which is the absolute term of the equation. 



Another Method. 



(a). First, suppose the two roots not equal to each other. 

 Call, for abbreviation, the two roots of the equation x^-\-ex-\-f=o 

 a and b. Then, by the definition of a root, we have 



a--\-ea-\-f=o (i) 



and b'-\-eb+f=o. (2) 



Subtracting (2) from (i) we obtain 



a^-—b'-{-e(a — b) = o, (2,) 



or, dividing through b}- a—b, 



a-^b+e=o, 

 or e= — (a + b). (4) 



That is, the coefficient of x is the sum of the roots with opposite 

 signs. 



Now substitute this value of <f in equation (1). It becomes 

 a^--a(a + b)±f=o, (s) 



or —ab+f=o, (6) 



whence f=<^b. (•]) 



That is, the absolute term is equal to the product of the two 

 roots. 



(b). If the two roots equal each other, that is, if each is equal 

 to a, the form (x—a)(x—b)=o becomes (x—a)(x—a)=o, or 



