Q) Let 'a' be a non zero real number & 
alpha comma beta space b e space t h e space r o o t s space o f space t h e space e q u a t i o n space a x squared plus 5 x plus 2 equals 0. space T h e n space t h e space a b s o l u t e space v a l u e space o f space t h e space d i f f e r e n c e space o f space t h e space r o o t s space o f space t h e space e q u a t i o n space
a cubed left parenthesis x plus 5 right parenthesis squared minus 25 a left parenthesis x plus 5 right parenthesis plus 50 equals 0 space i s spaceleft parenthesis A right parenthesis open vertical bar alpha squared minus beta squared close vertical bar space space space space space left parenthesis B right parenthesis open vertical bar alpha beta left parenthesis alpha squared minus beta squared right parenthesis close vertical bar space space space space space left parenthesis C right parenthesis open vertical bar fraction numerator alpha squared minus beta squared over denominator alpha beta end fraction close vertical bar space space space space left parenthesis D right parenthesis open vertical bar fraction numerator alpha squared minus beta squared over denominator alpha squared beta squared end fraction close vertical bar

Asked by araima2001 | 23rd Nov, 2016, 10:36: AM

Expert Answer:

begin mathsize 16px style straight alpha space and space straight beta space are space roots space of space the space equation space ax squared plus 5 straight x plus 2 space equals space 0.

straight alpha space plus space straight beta space space equals space fraction numerator negative 5 over denominator straight a end fraction
straight alpha space straight beta space space equals space 2 over straight a

Let space straight m space and space straight n space be space the space roots space of space space straight a cubed open parentheses straight x minus 5 close parentheses squared minus 25 straight a open parentheses straight x minus 5 close parentheses plus 50 equals 0.

straight m plus straight n space equals fraction numerator 25 straight a over denominator straight a cubed end fraction equals 25 over straight a squared equals open parentheses straight alpha space plus space straight beta close parentheses squared
mn space equals 50 over straight a cubed equals open parentheses straight alpha space plus space straight beta close parentheses squared straight alpha space straight beta space space

Now comma
open parentheses straight m minus straight n close parentheses squared equals open parentheses straight m plus straight n close parentheses squared minus 4 mn
space space space space space space space space space space space space space space space space equals open parentheses straight alpha space plus space straight beta close parentheses to the power of 4 minus 4 open parentheses straight alpha space plus space straight beta close parentheses squared straight alpha space straight beta space
space space space space space space space space space space space space space space space space equals open parentheses straight alpha space plus space straight beta close parentheses squared open curly brackets open parentheses straight alpha space plus space straight beta close parentheses squared minus 4 straight alpha space straight beta space close curly brackets
open parentheses straight m minus straight n close parentheses squared space space space equals open parentheses straight alpha space plus space straight beta close parentheses squared open curly brackets open parentheses straight alpha space minus space straight beta close parentheses squared space close curly brackets

Taking space square space root
open vertical bar straight m minus straight n close vertical bar space equals open vertical bar open parentheses straight alpha space plus space straight beta close parentheses open parentheses straight alpha space minus space straight beta close parentheses close vertical bar space equals open vertical bar straight alpha squared minus space straight beta squared close vertical bar

end style

Answered by Vijaykumar Wani | 23rd Nov, 2016, 04:49: PM

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