Question

Ma puteti ajuta ??????

Answer 1

Răspuns:

Explicație pas cu pas:

a)

$\dfrac{x^{2}-18x+81}{x^{2} -81} = \dfrac{x^{2}-2*9x+9^{2} }{x^{2} -81} = \dfrac{(x - 9)^{2}}{(x-9)(x+9)} = \dfrac{x - 9}{x+9}$

$x \in \mathbb{R} - \{-9;9\}$

b)

$\dfrac{x^{2}-36}{x^{2}-12x+36} = \dfrac{x^{2}-6^{2} }{x^{2}-2*6x+6^{2}} = \dfrac{(x-6)(x+6)}{(x-6)^{2}} = \dfrac{x+6}{x-6}$

$x \in \mathbb{R} - \{6\}$

c)

$\dfrac{x^{2}-64}{x^{2}+16x+64} = \dfrac{x^{2}-8^{2} }{x^{2}+2*8x+8^{2}} = \dfrac{(x-8)(x+8)}{(x+8)^{2}} = \dfrac{x-8}{x+8}$

$x \in \mathbb{R} - \{-8\}$

f)

$\dfrac{x^{4}-100x^{2}}{x^{3}+20x^{2}+100x} = \dfrac{x^{2}(x^{2}-10^{2}) }{x(x^{2}+2*10x+10^{2})} = \dfrac{x^{2}(x-10)(x+10)}{x(x+10)^{2}} = \dfrac{x(x-10)}{x+10}$

$x \in \mathbb{R} - \{-10;0\}$

Answer 2

Răspuns:

Explicație pas cu pas:

Domeniul de definitie il afli punand conditia ca numitorul fractiei sa fie diferit de 0.

a)

x^2 - 81 = 0

x^2 = 81

x1 = 9

x2 = -9

x ∈ R \ {-9, 9}

x^2 - 18x + 81 = (x - 9)^2

x^2 - 81 = (x + 9)(x - 9)

fractia devine

(x - 9)^2/(x + 9)(x - 9) = (x - 9)/(x + 9)

b)

x^2 - 12x + 36 = 0

(x - 6)^2 = 0

x = 6

x ∈ R \ {6}

x^2 - 36 = (x + 6)(x - 6)

fractia devine

(x + 6)(x - 6) / (x - 6)^2 = (x + 6)/(x - 6)

c)

x^2 + 16x + 64 = (x + 8)^2

(x + 8)^2 = 0

x = -8

x ∈ R \ {-8}

x^2 - 64 = (x + 8)(x - 8)

fractia devine

(x + 8)(x - 8) / (x + 8)^2  = (x - 8)/(x + 8)

d)

x^3 - 25x = 0

x*(x^2 - 25) = 0

x*(x + 5)*(x - 5) = 0

x ∈ R \ {-5; 0; 5}

x^2 + 10x + 25 = (x + 5)^2

fractia devine

(x + 5)^2 / x*(x + 5)*(x - 5) = (x + 5) / x*(x - 5)

e)

x^3 + 14x^2 + 49x = x*(x^2 + 14x + 49) = x*(x + 7)^2

x*(x + 7)^2 = 0

x ∈ R \ {-7; 0}

x^3 - 49x = x*(x^2 - 49) = x*(x + 7)*(x - 7)

fractia devine

x*(x + 7)*(x - 7) / x*(x + 7)^2 = (x - 7) / (x + 7)

f)

x^3 + 20x^2 + 100x = x*(x^2 + 20x + 100) = x*(x + 10)^2

x ∈ R \ {-10; 0}

x^4 - 100x^2 = x^2*(x^2 - 100) = x^2*(x - 10)(x + 10)

x^2*(x - 10)(x + 10) / x*(x + 10)^2 = x*(x - 10) / (x + 10)