Math, asked by gaurang5, 1 year ago

If x=7+√3 then find
x^2 + 1/x^2

Note :- please give some applicable method and not only calculations.

Answers

Answered by abhi569
4
Given that x =7 + 3

===================

1/x = 1/(7 + 
3)

By Rationalization,

 \frac{1}{x} =  \frac{1}{7 +  \sqrt{3} }  × \frac{7 -  \sqrt{3} }{7 - \sqrt{3} }

 \frac{1}{x} =  \frac{7 -  \sqrt{3} }{7^2 - ( \sqrt{3})^2 }

 \frac{1}{x} =   \frac{7 -  \sqrt{3} }{49 - 3 }

 \frac{1}{x } =   \frac{7 -  \sqrt{3} }{46}

===============================


Then,

x
² =  (7 + 3)²
    = 49 + 3 + 143
    = 52 + 14
3

AND

1/x
²  (\frac{7 -  \sqrt{3} }{46} )^2

1/x² =  \frac{49 + 3 - 14 \sqrt{3} }{2116}

1/x² =  \frac{52 -14 \sqrt{3} }{2116}


===========================

x² + 1/x²  = 52 + 14√3 + (52 - 14√3)/2116 

               = (110032+ 29624√3 + 52 - 14√3)/2116 
              
               = (110084 + 29610√3)/2116 



On Finding any mistake, comment

i hope this will help you



-by ABHAY

gaurang5: ok it was helpful but please can you provide some easier solution
Anonymous: your answer is little wrong
abhi569: wait
Anonymous: there are 110713 in place of 114316
abhi569: see it again
Anonymous: 110713 hoga
abhi569: please see it again and again
Anonymous: see my answer
Anonymous: how can i do
Anonymous: and then see your method
Answered by Anonymous
2
Hola !!

your answer is ---

Given,
x = 7 + \sqrt{3}

so ,

by rationalization
=============
 \frac{1}{x} = \frac{7 - \sqrt{3} }{(7 + \sqrt{3})(7 - \sqrt{3} } \\ = \frac{7 - \sqrt{3} }{ {7}^{2} - { \sqrt{3} }^{2} } = \frac{7 - \sqrt{3} }{46}

now ,
x + \frac{1}{x} = 7 + \sqrt{3} + \frac{7 - \sqrt{3} }{46} \\ = \frac{322 + 46 \sqrt{3} + 7 - \sqrt{3} }{46} \\ = \frac{329 + 45 \sqrt{3} }{46}

now,
 {x}^{2} + \frac{1}{ {x}^{2} } = {(x + \frac{1}{x}) }^{2} - 2 \\ = { \frac{(329 + 45 \sqrt{3)} }{2116}}^{2} - 2 \\ = \frac{108241 + 6075+ 29610 \sqrt{3} }{2116} - 2 \\ = \frac{110084<br /> + 29610 \sqrt{3} }{2116}
============

hope it help you

gaurang5: ok it was helpful but it busted my head
gaurang5: i undererood it but at competitive level i dont have time to calc. all the stuff
Anonymous: there is no formula for this calc
mysticd: You did a mistake , plz , edit
mysticd: ( 45√3 )² = 6075
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