Question

If  \[3x + \frac{2}{x} = 7\] , then \[\left( 9 x^2 - \frac{4}{x^2} \right) =\]

Options

• 25

• 35

• 49

• 30

Answer 1

We have to find the value of  `(9x^2 - 4/x^2)`

Given  `3x +2/x = 7`

Using identity  `(a+b)^2 = a^2 +b^2 +2ab`  we get,

Here ` a = 3x ,b= 2/x`

`(3x +2/x )^2 = (3x)^2 + 2 xx 3x xx 2/x + (2/x)^2`

Substituting  `3x + 2/x = 7` we get,

`(7)^2 = 9x^2 + 2 xx 3 xx x xx 2/x +(2/x)^2``

`49 = 9x^2 + 12 +4/x^2`

By transposing  + 12  left hand side we get,

`49 - 12 = 9x^2 +4/x^2`

         `37 = 9x^2 + 4/ x^2`

Again using identity   `(a-b)^2 = a^2 - 2ab +b^2` we get,

`(3x - 2/x)^2 = (3x )^2 - 2 xx 3x xx 2/x + (2/x)^2`

`(3x- 2/x)^2=(9x)^2 + 4/x^2 - 12`

Substituting  `(9x)^2 + 4/x^2 = 37` we get 

`(3x - 2/x)^2 = 37 - 12`

`(3x - 2/x)^2 = 25`

`(3x - 2/x)(3x - 2/x) = 5 xx 5`

       `3x - 2/x = 5`

Using identity  (x + y)( x - y )we get 

Here ` x= 3x,y = 2/x`

 `(3x)^2 - (2/x)^2 = (3x + 2/x)(3x - 2/x)`

Substituting  `3x +2/x = 7,3x - 2/x = 5` we get,

`9x^2 - 4/x^2 = 7 xx 5 `

`9x^2 - 4/x^2 = 35`

The value of  `9x^2 - 4/x^2`is 35.