Question

Mentionati conditiile in care fractiile date sunt definite si simplificati-le : a) x la a 2-a -4 / (x+2) la a 2-a b) (x-3)(x+3) / x la a 2-a - 6x + 9 c) x la a 2-a - 4 / x la a 2-a - 4x + 4 d) x la a treia - 9x / x la a treia - 6x la a doua + 9x e) 4x la a doua + 20x + 25 / 4x la a doua - 25 f) x la a patra - 16 / x la a patra - 8x la a doua + 16

Answer 1

$a) \frac{ {x}^{2} - 4 }{ {(x + 2)}^{2} } = \frac{ {x}^{2} - {2}^{2} }{ {(x + 2)}^{2} } = \frac{(x - 2)(x + 2)}{ {(x + 2)}^{2} } = \frac{x - 2}{ {(x + 2)}^{2 - 1} } = \frac{x - 2}{x + 2}$

$b) \frac{(x - 3)(x + 3)}{ {x}^{2} - 6x + 9 } = \frac{(x - 3)(x + 3)}{ {x}^{2} + 2x( - 3) + {( - 3)}^{2} } = \frac{(x - 3)(x + 3)}{ {(x - 3)}^{2} } = \frac{x + 3}{ {(x - 3)}^{2 - 1} } = \frac{x + 3}{x - 3}$

$c) \frac{ {x}^{2} - 4 }{ {x}^{2} - 4x + 4 } = \frac{ {x}^{2} - {2}^{2} }{ {x}^{2} + 2x( - 2) + {( - 2)}^{2} } = \frac{(x - 2)(x + 2)}{ {(x - 2)}^{2} } = \frac{x + 2}{ {(x - 2)}^{2 - 1} } = \frac{x + 2}{x - 2}$

$d) \frac{ {x}^{3} - 9x }{ {x}^{3} - {6x}^{2} + 9x } = \frac{ {x}^{3} - {3}^{2} \times x }{ {x}^{3} - 2 \times 3 \times {x}^{2} + {3}^{2} \times x } = \frac{x( \frac{ {x}^{3} }{x} - \frac{ {3}^{2} \times x }{x}) }{x( \frac{ {x}^{3} }{x} - \frac{2 \times {3x}^{2} }{x} + \frac{ {3}^{2} \times x }{x}) } = \frac{x( {x}^{3 - 1} - {(3}^{2})) }{x( {x}^{3 - 1} - (2 \times {3x}^{2 - 1}) + {3}^{2}) } = \frac{x( {x}^{2} - 9) }{x( {x}^{2} - (2 \times 3x) + 9) } = \frac{x( {x}^{2} - 9) }{x( {x}^{2} - (6x) + 9) } = \frac{ {x}^{2} - 9 }{ {x}^{2} - 6x + 9 } = \frac{ {x}^{2} - {3}^{2} }{ {x}^{2} + 2x( - 3) + {( - 3)}^{2} } = \frac{(x - 3)(x + 3)}{ {(x - 3)}^{2} } = \frac{x + 3}{ {(x - 3)}^{2 - 1} } = \frac{x + 3}{x - 3}$

$e) \frac{ {4x}^{2} + 20x + 25 }{ {4x}^{2} - 25 } = \frac{ {(2x)}^{2} + 2(2x)5 + {5}^{2} }{ {2}^{2} \times {x}^{2} - {5}^{2} } = \frac{ {(2x + 5)}^{2} }{ {(2x)}^{2} - {5}^{2} } = \frac{ {(2x + 5)}^{2} }{(2x - 5)(2x + 5)} = \frac{ {(2x + 5)}^{2 - 1} }{2x - 5} = \frac{2x + 5}{2x - 5}$

$f) \frac{ {x}^{4} - 16 }{ {x}^{4} - {8x}^{2} + 16 } = \frac{ {x}^{4} - {2}^{4} }{ { {(x}^{2}) }^{2} + 2 {(x}^{2})( - 4) + {( - 4)}^{2} } = \frac{ { {(x}^{2}) }^{2} - { {(2}^{2}) }^{2} }{ { {(x}^{2} - 4) }^{2} } = \frac{( {x}^{2} - {2}^{2})( {x}^{2} + {2}^{2}) }{ { {(x}^{2} - {2}^{2}) }^{2} } = \frac{( {x}^{2} - 4)( {x}^{2} + 4) }{((x - 2)(x + 2))^{2} } = \frac{( {x}^{2} - 4)( {x}^{2} + 4) }{ {(x - 2)}^{2} {(x + 2)}^{2} } = \frac{( {x}^{2} - {2}^{2})( {x}^{2} + 4) }{ {(x - 2)}^{2} {(x + 2)}^{2} } = \frac{((x - 2)(x + 2))( {x}^{2} + 4) }{ {(x - 2)}^{2} {(x + 2)}^{2} } = \frac{(x - 2)(x + 2)( {x}^{2} + 4) }{ {(x - 2)}^{2} {(x + 2)}^{2} } = \frac{ {x}^{2} + 4 }{ {(x - 2)}^{2 - 1} {(x + 2)}^{2 - 1} } = \frac{ {x}^{2} + 4 }{(x - 2)(x + 2)} = \frac{ {x}^{2}+4 }{ {x}^{2} - 4}$