Question

Solve:

`9(x^2 + 1/x^2) - 9(x + 1/x) - 52 = 0`

Answer 1

Given equation be,

`9(x^2 + 1/x^2) - 9(x + 1/x) - 52 = 0`

Put `x + 1/x = y`

Then `x^2 + 1/x^2 + 2 = y^2`  ...[On squaring]

`\implies x^2 + 1/x^2 = y^2 - 2`

Thus given equation

∴ `9(x^2 + 1/x^2) - 9(x + 1/x) - 52 = 0`

`\implies` 9(y² – 2) – 9y – 52 = 0

`\implies` 9y² – 18 – 9y – 52 = 0

`\implies` 9y² – 9y – 70 = 0

`\implies` 9y² – 30y + 21y – 70 = 0

`\implies` 3y(3y – 10) + 7(3y – 10) = 0

`\implies` (3y – 10)(3y + 7) = 0 

Either 3y – 10 = 0, then y = `10/3`

Or 3y + 7 = 0 then y = `(-7)/3`

(i) When y = `10/3`, then `x + 1/x = 10/3`

`\implies` 3x² + 3 = 10x

`\implies` 3x² – 10x + 3 = 0

`\implies` 3x² – 9x – x + 3 = 0

`\implies` 3x(x – 3) – 1(x – 3) = 0

`\implies` (x – 3)(3x – 1) = 0

Either x – 3 = 0, then x = 3

Or 3x – 1 = 0, then x = `1/3`

(ii) When `y = (-7)/3`, then `x + 1/x = (-7)/3`

`\implies` 3x² + 3 = –7x

`\implies` 3x² + 7x + 3 = 0

Here a = 3, b = 7, c = 3

∴ D = b² – 4ac

= (7)² – 4 × 3 × 3

= 49 – 36

= 13

∴ `x = (-b xx sqrt(D))/(2a)`

= `(-7 +- sqrt(13))/(2 xx 3)`

= `(-7 +- sqrt(13))/6`

∴ x = `3, 1/3, (-7 +- sqrt(13))/6`