Find the value of alpha to the power of negative 1 end exponent and beta to the power of negative 1 end exponent, if alpha and beta are the zeroes of x squared minus 4 x plus 1.

Asked by Jerlin George | 2nd Oct, 2015, 02:41: PM

Expert Answer:

C o n s i d e r space t h e space q u a d r a t i c space e q u a t i o n space x squared minus 4 x plus 1 G i v e n space t h a t space alpha space a n d space beta space a r e space t h e space r o o t s space o f space t h e space a b o v e space e q u a t i o n. T h u s space x equals fraction numerator negative open parentheses negative 4 close parentheses plus-or-minus square root of open parentheses negative 4 close parentheses squared minus 4 end root over denominator 2 end fraction rightwards double arrow space x equals fraction numerator 4 plus-or-minus square root of 16 minus 4 end root over denominator 2 end fraction rightwards double arrow space x equals fraction numerator 4 plus-or-minus square root of 12 over denominator 2 end fraction rightwards double arrow space x equals fraction numerator 4 plus-or-minus 2 square root of 3 over denominator 2 end fraction rightwards double arrow space x equals 2 plus-or-minus square root of 3 rightwards double arrow alpha equals 2 plus square root of 3 space a n d space beta equals 2 minus square root of 3 W e space n e e d space t o space f i n d space t h e space v a l u e space o f space 1 over alpha space a n d space 1 over beta colon 1 over alpha equals fraction numerator 1 over denominator 2 plus square root of 3 end fraction cross times fraction numerator 2 minus square root of 3 over denominator 2 minus square root of 3 end fraction space space space space equals fraction numerator 2 minus square root of 3 over denominator 2 squared minus open parentheses square root of 3 close parentheses squared end fraction space space space space equals fraction numerator 2 minus square root of 3 over denominator 4 minus 3 end fraction space space space space equals 2 minus square root of 3  1 over beta equals fraction numerator 1 over denominator 2 minus square root of 3 end fraction cross times fraction numerator 2 plus square root of 3 over denominator 2 plus square root of 3 end fraction space space space space equals fraction numerator 2 plus square root of 3 over denominator 2 squared minus open parentheses square root of 3 close parentheses squared end fraction space space space space equals fraction numerator 2 plus square root of 3 over denominator 4 minus 3 end fraction space space space space equals 2 plus square root of 3  T h u s space t h e space v a l u e s space o f space 1 over alpha space a n d space 1 over beta space a r e colon space 2 minus square root of 3 space a n d space 2 plus square root of 3 space

Answered by Vimala Ramamurthy | 2nd Oct, 2015, 10:54: PM

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