Question

If x^2 + 1/x^2=23 find the value of (x^2-1/x^2)​

Answer 1

Answer:

Step-by-step explanation:

x^4 + 1 = 23x^2

x^4 - 23x^2 + 1 = 0

let x^2 = y

y^2 -23y + 1 = 0

Applying quadratic formula gives:

y = (23 - 5√21)/2 and y = (23 + 5√21)/2

y = x^2

so x^2 = (23 - 5√21)/2 and x^2 = (23 + 5√21)/2

x^2 - 1/x^2 = (23 - 5√21)/2 - 2/(23 - 5√21) = [(23 - 5√21)^2 - 2(23 - 5√21)]/2(23 - 5√21) = (23 - 5√21)[(23 - 5√21) - 2]/2(23 - 5√21) = (21 - 5√21)/2

and

x^2 - 1/x^2 = (23 + 5√21)/2 - 2/(23 + 5√21) = [(23 + 5√21)^2 - 2(23 + 5√21)]/2(23 + 5√21) = (23 + 5√21)[(23 + 5√21) - 2]/2(23 + 5√21) = (21 + 5√21)/2