Question

Se stie ca x-1/x=3.Calculati: a) x²+1/x² ;b) x³-1/x³ ; c) x⁴+1/x⁴

Answer 1

Răspuns:

Explicație pas cu pas:

x - 1/x = 3  

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a)  

(x - 1/x)² = 3²

->  aplic formula ( a - b)² = a² - 2ab + b²; unde a = x, iar b = 1/x

x² - 2 × x × 1/x + 1/x² = 9

      2× (x/x) = 2 × 1 = 2

x² - 2 + 1/x² = 9

x² + 1/x² = 9 + 2

x² + 1/x² = 11

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b)

( x - 1/x)³ = 3³

x³ - 1/x³ - 3 ×x×1/x ( x - 1/x) = 27

x³ - 1/x³ - 3 × 3 = 27

x³ - 1/x³ = 27 + 9

x³ - 1/x³ = 36

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c)

( x - 1/x)² = 3²

x² + 1/x² = 11  -> ridic relatia din nou la patrat

(x² + 1/x²)² = 11²  

x⁴ + 1/x⁴+ 2x²/x² = 121

                2x²/x² = 2

x⁴ + 1/x⁴ = 121 - 2

x⁴ + 1/x⁴ = 119