Question

if x^2-5x+1=0 find the value of x^4 -1/x^4

Answer 1

$\huge\boxed{Answer}$

x² - 5x + 1 = 0

x² + 1 = 5x

x + 1/x = 5 -------(1)

Doing square both side

x² + 1/x² + 2 = 25

x² + 1/x² = 23 -----(2)

&

x² + 1/x² - 2 = 21

(x - 1/x)² = 21

(x - 1/x) = √21 ------(3)

From eq.(1) × eq.(3) -

(x + 1/x) (x - 1/x) = 5√21

x² - 1/x² = 5√21 ------(4)

Now ,

From eq.(2) × eq.(4) -

(x² + 1/x²) (x² - 1/x²) = 23*(5√21)

x⁴ - 1/x⁴ = 115√21