Find the values of a and b if:

$fraction numerator 7 plus 3 square root of 5 over denominator 3 plus square root of 5 end fraction minus fraction numerator 7 minus 3 square root of 5 over denominator 3 minus square root of 5 end fraction equals straight a plus square root of 5 straight b.$

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Solution

fraction numerator 7 plus 3 square root of 5 over denominator 3 plus square root of 5 end fraction minus fraction numerator 7 minus 3 square root of 5 over denominator 3 minus square root of 5 end fraction equals straight a plus square root of 5 straight b. rightwards double arrow space space space fraction numerator left parenthesis 7 plus 3 square root of 5 right parenthesis left parenthesis 3 minus square root of 5 right parenthesis over denominator left parenthesis 3 plus square root of 5 right parenthesis left parenthesis 3 minus square root of 5 right parenthesis end fraction minus fraction numerator left parenthesis 7 minus 3 square root of 5 right parenthesis left parenthesis 3 plus square root of 5 right parenthesis over denominator left parenthesis 3 minus square root of 5 right parenthesis left parenthesis 3 plus square root of 5 right parenthesis end fraction equals straight a plus square root of 5 straight b. rightwards double arrow space space space space space space fraction numerator 21 minus 7 square root of 5 plus 9 square root of 5 minus 15 over denominator 9 minus 5 end fraction minus fraction numerator 21 plus 7 square root of 5 minus 9 square root of 5 minus 15 over denominator 9 minus 5 end fraction equals space 1 space plus space square root of 5 space b rightwards double arrow space fraction numerator 6 plus 2 square root of 5 over denominator 4 end fraction minus fraction numerator 6 minus 2 square root of 5 over denominator 4 end fraction equals a plus square root of 5 b rightwards double arrow space space fraction numerator left parenthesis 6 plus 2 square root of 5 right parenthesis minus left parenthesis 6 minus 2 square root of 5 right parenthesis over denominator 4 end fraction equals a plus square root of 5 b rightwards double arrow space space fraction numerator 4 square root of 5 over denominator 4 end fraction equals a plus square root of 5 b rightwards double arrow space space space square root of 5 space equals a plus square root of 5 b rightwards double arrow space a equals 0 comma space b space equals space 1

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Polynomials

Find the values of a and b if:

$fraction numerator 7 plus 3 square root of 5 over denominator 3 plus square root of 5 end fraction minus fraction numerator 7 minus 3 square root of 5 over denominator 3 minus square root of 5 end fraction equals straight a plus square root of 5 straight b.$

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