If x = √3+1 /2, then the value of x2  + 1/x2 and x2  –  1/x2

Asked by Archeet | 2nd Jun, 2014, 09:59: AM

Expert Answer:

G i v e n space t h a t space x equals fraction numerator square root of 3 plus 1 over denominator 2 end fraction ; L e t space u s space f i n d space 1 over x : 1 over x equals fraction numerator 1 over denominator fraction numerator square root of 3 plus 1 over denominator 2 end fraction end fraction rightwards double arrow 1 over x equals fraction numerator 2 over denominator square root of 3 plus 1 end fraction rightwards double arrow 1 over x equals fraction numerator 2 over denominator square root of 3 plus 1 end fraction cross times fraction numerator square root of 3 minus 1 over denominator square root of 3 minus 1 end fraction rightwards double arrow 1 over x equals fraction numerator 2 open parentheses square root of 3 minus 1 close parentheses over denominator 3 minus 1 end fraction rightwards double arrow 1 over x equals square root of 3 minus 1 T h e n comma space x plus 1 over x equals fraction numerator square root of 3 plus 1 over denominator 2 end fraction plus square root of 3 minus 1 rightwards double arrow space x plus 1 over x equals fraction numerator open parentheses square root of 3 plus 1 close parentheses plus 2 open parentheses square root of 3 minus 1 close parentheses over denominator 2 end fraction rightwards double arrow space x plus 1 over x equals fraction numerator open parentheses square root of 3 plus 1 close parentheses plus 2 square root of 3 minus 2 over denominator 2 end fraction rightwards double arrow space x plus 1 over x equals fraction numerator 3 square root of 3 minus 1 over denominator 2 end fraction S i m i l a r l y comma space x minus 1 over x equals fraction numerator square root of 3 plus 1 over denominator 2 end fraction minus open parentheses square root of 3 minus 1 close parentheses rightwards double arrow space x minus 1 over x equals fraction numerator open parentheses square root of 3 plus 1 close parentheses minus 2 open parentheses square root of 3 minus 1 close parentheses over denominator 2 end fraction rightwards double arrow space x minus 1 over x equals fraction numerator open parentheses square root of 3 plus 1 close parentheses minus 2 square root of 3 plus 2 over denominator 2 end fraction rightwards double arrow space x minus 1 over x equals fraction numerator minus square root of 3 plus 3 over denominator 2 end fraction T h u s comma space open parentheses space x plus 1 over x space close parentheses open parentheses space x minus 1 over x close parentheses equals open parentheses fraction numerator 3 square root of 3 minus 1 over denominator 2 end fraction close parentheses cross times open parentheses fraction numerator minus square root of 3 plus 3 over denominator 2 end fraction close parentheses rightwards double arrow x squared minus 1 over x squared equals fraction numerator minus 12 plus 10 square root of 3 over denominator 4 end fraction equals fraction numerator minus 6 plus 5 square root of 3 over denominator 2 end fraction
N o w space c o n s i d e r space open parentheses x plus 1 over x close parentheses squared : open parentheses x plus 1 over x close parentheses squared equals x squared plus 1 over x squared plus 2 rightwards double arrow x squared plus 1 over x squared equals open parentheses x plus 1 over x close parentheses squared minus 2 rightwards double arrow x squared plus 1 over x squared equals open square brackets fraction numerator 3 square root of 3 minus 1 over denominator 2 end fraction close square brackets squared minus 2 rightwards double arrow x squared plus 1 over x squared equals open square brackets fraction numerator 9 cross times 3 plus 1 minus 6 square root of 3 over denominator 4 end fraction close square brackets minus 2 rightwards double arrow x squared plus 1 over x squared equals fraction numerator 28 minus 6 square root of 3 minus 8 over denominator 4 end fraction rightwards double arrow x squared plus 1 over x squared equals fraction numerator 20 minus 6 square root of 3 over denominator 4 end fraction rightwards double arrow x squared plus 1 over x squared equals fraction numerator 10 minus 3 square root of 3 over denominator 2 end fraction

Answered by Vimala Ramamurthy | 2nd Jun, 2014, 12:56: PM