Question

plz answer this maths problem step by step ​

Answer 1

Question:

If x² - 1/x² = 2, then find x² + 1/x² and x + 1/x.

Solution:

x² - 1/x² = 2

=> x² + (- 1/x²) = 2

=> x² + (i/x)² = 2

x² + (i/x)² + (2 · x · i/x) = 2 + (2 · x · i/x))

=> (x + i/x)² = 2 + 2i

=> x + i/x = √(2 + 2i) → (1)

x² + (i/x)² - (2 · x · i/x) = 2 - (2 · x · i/x))

=> (x - i/x)² = 2 - 2i

=> x - i/x = √(2 - 2i) → (2)

Multiplying (1) and (2),

(x + i/x)(x - i/x) = (√(2 + 2i))(√(2 - 2i))

=> x² - (- 1/x²) = √[(2 + 2i)(2 - 2i)]

=> x² + 1/x² = √[4 - (- 4)]

=> x² + 1/x² = √8

=> x² + 1/x² = 2√2

x² + 1/x² + 2 = 2√2 + 2

=> (x + 1/x)² = 2(√2 + 1)

=> x + 1/x = √[2(√2 + 1)]

=> x + 1/x = √2 · √(√2 + 1)

Hence found!

Note that i = √(-1).