Question

If x² + `x^(1/2)`= 7 and  x ≠ 0; find the value of: 
7x³ + 8x − `7/x^3 - 8/x`

Answer 1

7x³ + 8x − `7/x^3 - 8/x`

terms with common factors: `7(x^3 - 1/x^3) + 8 (c-1/x)`

Finding the value of `(x-1/x)`

`(x-1/x)^2 = x^2 + 1/x^2 -2`

`(x-1/x)^2 = 7-2=5`

Therefore, `x-1/x = +-sqrt5`

`(x-1/x)^3 = x^3-1/x^3-3(x-1/x)`

`x^3-1/x^3 = (x-1/x)^3 +3(x-1/x)`

`x-1/x = +-sqrt5: x^3-1/x^3 = (+-sqrt5)^3 + 3(+-sqrt5)`

`=x^3 -1/x^3=+-5sqrt5 +-3sqrt5.x^3-1/x^3 = +-8sqrt5`

The values found for `(x-1/x) and (x^3-1/x^3)` are substituted into the expression from step `1:7 (+-8sqrt5) +8 (+-sqrt5).`

`7x^3+8x-7/x^3-8/x is +-64sqrt5`