86 Algebra. 



16. TheorKM. .-///_>' equatio7i "cvhich can be placed in the form 

 x^"-\-ex"-\-f=^o can be solved as a quadratic. 



may be written 



which, if we regard (x") as the unknown qantity, is seen to be in 

 the quadratic form. Completing the square of (2) it becomes 



(x'^:^Jre(xn -f \e^^Y-f (i) 



Whence x" + i^'= i V \(f—f 



or X" = — }e± \/ \(^ —J. 



Therefore x=( —^^e±i,\/ \e^—f)" (4) 



which is the sohition of the equation of the proposed form. 



17. Theorem. Any equation ivhicli can be placed in the form 

 X' -\-eX''-\-f=o, ivhere X stands for any lifiear or quadratic, 

 function of the unhioivn, can be solved as a quadratic. 



For, by the last article, it will be found that 



1 



Now, if X is a linear function 0/ x, this equation is of the form 



1 



ax-^b={-\e±^J\c-fY (^) 



which can be easily solved for .r. 



If X\s a quadratic function of .r equation (\) must be of the 



form 



1 



ax^^bx^-c={^-\e^s/\e^-fY (2>) 



Now this is a quadratic equation in terms of x^ since all 

 other quantities in the equation are known, and hence the equa- 

 tion can be solved. 



In treating examples which come under these two theorems it 

 may be possible that we will not find all the values that will sat- 

 isfy the given equation. This happens because we are not always 

 able to find n different n th roots of a quantity, while that num- 

 ber really do exist. Thus from the equation 



jt-H 19.^3=216 (\) 



we will find by considering x'^ the unknown quantity that 



Jf3=27 or —8 

 whence -^=3 or —2. 



