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Polynomials

Question

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If $straight a equals 7 minus 4 square root of 3 comma$ find the value of $square root of straight a plus fraction numerator 1 over denominator square root of a end fraction.$

Solution

straight a equals 7 minus 4 square root of 3 equals 4 plus 3 minus 4 square root of 3 equals 2 squared plus left parenthesis square root of 3 right parenthesis squared minus 2 left parenthesis 2 right parenthesis left parenthesis square root of 3 right parenthesis equals space left parenthesis 2 minus square root of 3 right parenthesis squared rightwards double arrow space space space square root of straight a space equals 2 minus square root of 3 space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 1 right parenthesis space space space space space space fraction numerator 1 over denominator square root of straight a end fraction equals fraction numerator 1 over denominator 2 minus square root of 3 end fraction space space space space space equals space 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 space equals space fraction numerator 2 plus square root of 3 over denominator 4 minus 3 end fraction space space space space space equals space 2 plus square root of 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 2 right parenthesis

From (1) and (2),

square root of straight a plus fraction numerator 1 over denominator square root of a end fraction equals left parenthesis 2 minus square root of 3 right parenthesis plus left parenthesis 2 plus square root of 3 right parenthesis equals 4

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