if x + 1/x = 6 , find the value of: a) x² + 1/x² b) x⁴ - 1/x⁴
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a) x² + 1/x²
(x + 1/x)² = x² + 2 + 1/x²
36 = x² + 2 + 1/x² → ∴x² + 1/x² = 34
b) x⁴ - 1/x⁴
We will find x² - 1/x² then multiply by x² + 1/x².
(x - 1/x)² = x² - 2 + 1/x² → (x - 1/x)² = 32
∴x - 1/x = ±4√2
∴x² - 1/x² = ±24√2
∴x⁴ - 1/x⁴ = ±144√2
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