If X=[(a+2b)^1/2+(a-2b)^1/2] / [(a+2b)^1/2-(a-2b)^1/2] then prove (b^2*x^2)-abx+b^2=0
 

Asked by abinash.gupta003 | 24th Jul, 2016, 02:30: PM

Expert Answer:

begin mathsize 12px style Given space straight x equals fraction numerator square root of straight a plus 2 straight b end root plus square root of straight a minus 2 straight b end root over denominator square root of straight a plus 2 straight b end root minus square root of straight a minus 2 straight b end root end fraction
Rationalise space the space denominator
straight x equals fraction numerator left parenthesis square root of straight a plus 2 straight b end root plus square root of straight a minus 2 straight b end root right parenthesis left parenthesis square root of straight a plus 2 straight b end root plus square root of straight a minus 2 straight b end root right parenthesis over denominator left parenthesis square root of straight a plus 2 straight b end root minus square root of straight a minus 2 straight b end root right parenthesis left parenthesis square root of straight a plus 2 straight b end root plus square root of straight a minus 2 straight b end root right parenthesis end fraction
rightwards double arrow straight x equals fraction numerator left parenthesis square root of straight a plus 2 straight b end root plus square root of straight a minus 2 straight b end root right parenthesis squared over denominator straight a plus 2 straight b minus straight a plus 2 straight b end fraction
rightwards double arrow straight x equals fraction numerator straight a plus 2 straight b plus straight a minus 2 straight b plus 2 square root of straight a plus 2 straight b end root square root of straight a minus 2 straight b end root over denominator 4 straight b end fraction
rightwards double arrow straight x equals fraction numerator 2 straight a plus 2 square root of straight a plus 2 straight b end root square root of straight a minus 2 straight b end root over denominator 4 straight b end fraction
rightwards double arrow straight x equals fraction numerator straight a plus square root of straight a plus 2 straight b end root square root of straight a minus 2 straight b end root over denominator 2 straight b end fraction
rightwards double arrow 2 bx equals straight a plus square root of straight a plus 2 straight b end root square root of straight a minus 2 straight b end root
rightwards double arrow 2 bx minus straight a equals square root of straight a plus 2 straight b end root square root of straight a minus 2 straight b end root
Squaring space both space sides
rightwards double arrow left parenthesis 2 bx minus straight a right parenthesis squared equals left parenthesis square root of straight a plus 2 straight b end root square root of straight a minus 2 straight b end root right parenthesis squared
rightwards double arrow 4 straight b squared straight x squared plus straight a squared minus 4 abx equals left parenthesis straight a plus 2 straight b right parenthesis left parenthesis straight a minus 2 straight b right parenthesis
rightwards double arrow 4 straight b squared straight x squared plus straight a squared minus 4 abx equals straight a squared minus 4 straight b squared
rightwards double arrow 4 straight b squared straight x squared minus 4 abx equals negative 4 straight b squared
rightwards double arrow 4 straight b squared straight x squared minus 4 abx plus 4 straight b squared equals 0
rightwards double arrow straight b squared straight x squared minus abx plus straight b squared equals 0
Hence space proved. end style

Answered by Rebecca Fernandes | 25th Jul, 2016, 09:35: AM

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