Question

if x4+1/x4=194findx3+1/x3,x2+1/x2and x+1/x​

Answer 1

Given : x⁴ + 1/x⁴ = 194

To find : (x³ + 1/x³), (x² + 1/x²), (x + 1/x)

Solution :

Given, x⁴ + 1/x⁴ = 194

⇒ (x² + 1/x²)² - (2 * x² * 1/x²) = 194

⇒ (x² + 1/x²)² - 2 = 194

⇒ (x² + 1/x²)² = 194 + 2 = 196

⇒ x² + 1/x² = ± 14

Taking the + value only, we get

(x + 1/x)² - (2 * x * 1/x) = 14

⇒ (x + 1/x)² - 2 = 14

⇒ (x + 1/x)² = 14 + 2 = 16

⇒ x + 1/x = ± 4

Taking the - value only, we get

(x + 1/x)² - (2 * x * 1/x) = - 14

⇒ (x + 1/x)² - 2 = - 14

⇒ (x + 1/x)² = - 14 + 2 = - 12

⇒ x + 1/x = ± √(- 12)

⇒ x + 1/x = ± 2√3 i

{ For the computation of (x³ + 1/x³), we take the positive values only as in lower classes, complex numbers isn't in syllabus. }

Now, x³ + 1/x³

= (x + 1/x) (x² + 1/x² - 1)

= 4 (14 - 1)

= 4 * 13

= 52

⇒ x³ + 1/x³ = 52