Question

X^2 +x -1 = 0 what is value of x^4 + 1/x^4 gate

Answer 1

Answer:

7

Step-by-step explanation:

x² + x - 1 = 0

a = 1 , b = 1 , c = -1

x = -b ± √b² - 4ac/2a = -1 ±√1 + 4/2 = - 1 ±√5/2

if x = -1 + √5/2, 1/x = 2/√5 - 1   = 2 (√5 + 1)/(√5 - 1)(√5 + 1) = √5 + 1 / 2.

if x =  - 1 - √5/2, 1/x = -2/(√5 + 1) = - 2(√5 - 1)/(√5 - 1)(√5 + 1) = 1 - √5 / 2.

When x = √5 - 1/2 and 1/x = √5 + 1 / 2.

x² + 1/x² = (x + 1/x)² - 2 = (√5)² - 2 = 3.

x⁴ + 1/x⁴ = (x² + 1/x²)² - 2 = (3)² - 2 = 7.

When x = -1 - √5 /2 and 1/x = 1 - √5 / 2.

x² + 1/x² = (x + 1/x)² - 2 = (-√5)² - 2 = 3.

x⁴ + 1/x⁴ = (x² + 1/x²)² - 2 = (3)² - 2 = 7.

So by either value of x, x⁴ + 1/x⁴ = 7.