Question

Ex 24 26 și 27 va rog dau coroana și inima

Answer 1

Răspuns:

Explicație pas cu pas:

24) x + 1/x = 2

a) (x+1/x)² = 4 = x²+ 2·x·1/x + 1/x²  =>

x²+ 1/x² = 4-2 = 2

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b) x³+1/x³ = ?

(x+1/x)³ = (x+1/x)(x²+2·x·1/x+1/x²) =2³ = 8

(x+1/x)³ = x³+2x+1/x+x+2/x+1/x³ = 8 =>

x³+1/x³ = 8-(3x+3/x) = 8-6 = 2

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c) x⁴+1/x⁴ = ?

(x+1/x)⁴ = (x²+2·x·1/x+1/x²)·(x²+2·x·1/x+1/x²) = 2⁴ = 16

(x+1/x)⁴ = (x²+1/x²+2)· (x²+1/x²+2) = 16

(x+1/x)⁴ = x⁴+1+2x²+1+1/x⁴+2/x²+2x²+2/x²+4 = 16

x⁴+1/x⁴ = 16-(6+4x²+4/x²) = 16-(6+4·2) = 2

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26) x+y = 4 ; xy = 3 ; x,y ∈ R*

a) (x+y)² = 4² = 16

b) x²+y² = (x+y)²-2xy = 16-2·3 = 10

c) (x-y)² = (x²+y²)-2xy = 10-2·3 = 4

d) 1/x + 1/y = (y+x)/xy = 4/3

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27) a²+1/a² = 23

a) a+1/a ; a < 0

(a+1/a)² = a²+1/a²+2·a·1/a = 23+2 = 25 =>

a+1/a = -√25 = -5

b) (a-1/a) ; a > 1

(a-1/a)² = a²+1/a²-2·a·1/a = 23-2 = 21 =>

a-1/a = √21