Question

If x=2+√5 find the value of x²-1/x²

Answer 1

`{ \tt{ {x}^(2) - \frac{1}{ {x}^(2) } }} \\ = { \tt{ {(2 + √(5) )}^(2) - \frac{1}{ {(2 + √(5)) }^(2) } }} \\ = { \tt{ \frac{(2 + √(5) ) {}^(4) - 1}{ {(2 + √(5) )}^(2) } }} \\ = { \tt{ \frac{(9 + 4 √(5)) {}^(2) }{ {(9 + 4√(5)) }}}} \\ = { \tt{9 + 4 √(5) }}`

Answer 2

`8√(5)`

Explanation:

`x = 2 + √(5)\\\\ x^(2) = (2+ √(5))^(2) \\\\ \ \ \ \ = 2^(2)+2* √(5)*2+( √(5))^(2)\\\\`

`= 4 + 4 √(5)+5\\\\= 9+4 √(5)`

`(1)/(x^(2))=(1)/(9+4√(5))\\\\=(1*(9-4√(5))/((9+4√(5))(9-4√(5)))\\\\=(9-4√(5))/(9^(2)-(4√(5))^(2))\\\\=(9-4√(5))/(81-4^(2)(√(5))^(2))\\\\=(9-4√(5))/(81-16*5)\\\\=(9-4√(5))/(81-80)\\\\=(9-4√(5))/(1)\\\\=9-4√(5)`

`x^(2)-(1)/(x^(2))= 9 + 4√(5) -(9 - 4√(5))\\\\`

`= 9 + 4√(5) - 9 + 4√(5)\\\\= 9 - 9 + 4√(5) + 4√(5)\\\\= 8√(5)`