03d. Ecuaciones racionales ? ECUACIONES 1 BCT Luisa Muoz 1 ECUACIONES RACIONALES 1) 3 1 3 2 = + x x x(x 1) + 6 = 6(x 1) x 2 x + 6 = 6x 6 x 2 7x + 12 = 0

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  • ECUACIONES 1 BCT

    Luisa Muoz 1

    ECUACIONES RACIONALES

    1) 31

    3

    2=

    +

    x

    x

    x(x 1) + 6 = 6(x 1) x2 x + 6 = 6x 6 x2 7x + 12 = 0

    2 2 1

    3 2

    7 1x 4 x 4

    7 7 4112 7 1 2x7 121 2

    x 3 x 32

    + = = = = = = = = =

    2) 41

    81

    =+

    x

    xx

    (x + 1)(x 1) = 8x 4(x 1) x2 1 = 8x 4x + 4 x2 4x 5 = 0

    2 2 1

    3 2

    4 6x 5 x 5

    4 4 41( 5) 4 6 2x4 621 2

    x 1 x 12

    + = = = = = = = = =

    3) 2

    21

    2

    2 ++=

    x

    x

    x

    4x = 2(x 2) + (x + 2)(x 2) 4x = 2x 4 + x2 4 x2 2x 8 = 0

    2 2 1

    3 2

    2 6x 4 x 4

    2 2 41( 8) 2 6 2x2 621 2

    x 2 x 22

    + = = = = = = = = =

    4) x

    x 2 2x 1

    + = +

    (x + 2)(x 1) = x + 2(x 1) x2 x + 2x 2 = x + 2x 2 x2 2x = 0 x(x 2) = 01

    2

    x 0

    x 2 0 x 2

    =

    = =

    5) 016x

    64x

    22 =+

    x4 16x2 + 64 = 0 y2 16y + 64 = 0 (y 8)2 = 0 2y 8 x 8 x 2= = =

    6) 54= x

    432 + x

    22

    x4 45x2 + 324 = 0 y2 45y + 324 = 0

    2 1 1

    2 2

    45 27y 36 x 6

    45 45 41324 45 27 2y45 2721 2

    y 9 x 32

    + = = = = = = = =

  • ECUACIONES 1 BCT

    Luisa Muoz 2

    7) 2

    2

    x

    225x34 =

    34x2 x4 = 225 x4 34x2 + 225 = 0 y2 34y + 225 = 0

    2 1 1

    2 2

    34 16y 25 x 5

    34 45 41( 225) 34 16 2y34 1621 2

    y 9 x 32

    + = = = = = = = =

    8) 1x

    12x

    22

    +=

    x4 + x2 12 = 0 y2 + y 12 = 0

    2 1 1

    2

    1 7y 3 x 3

    1 ( 1) 41( 12) 1 7 2y1 721 2

    y 42

    + = = = = = = =

    9) 72

    16 x = 9 x

    2 2

    2

    144 = (x2 16)(x2 9) 144 = x4 9x2 16x2 + 144 x4 25x2 = 0 x2(x2 25) = 0

    21

    22

    x 0 x 0

    x 25 0 x 5

    = =

    = =

    10) 9 x

    28 =

    4

    32x2

    2

    -112 = (x2 32)(x2 9) -112 = x4 32x2 9x2 + 288 x4 41x2 + 400 = 0 y2 41 y + 400 = 0

    2 1 1

    2 2

    41 9y 25 x 5

    41 41 41400 41 9 2y41 921 2

    y 16 x 42

    + = = = = = = = =

    11) 1

    )1(5

    1

    12

    +

    =+

    x

    x

    x

    x

    (2x + 1)(x + 1) = (5x 5)(x 1) 2x2 + 3x + 1 = 5x2 10x + 5 3x2 13x + 4 = 0

    1 12

    2 2

    13 11x 4 x 4

    613 13 434 13 11x

    13 11 2 123 6x x

    6 6 3

    + = = = = = = = =

  • ECUACIONES 1 BCT

    Luisa Muoz 3

    12) 1

    1

    1

    12

    +=

    +

    x

    x

    x

    x

    (2x 1)(x 1) = (x + 1)2 2x2 2x x + 1 = x2 + 2x + 1 x2 5x = 0 x(x 5) = 0 1

    2

    x 0

    x 5 0 x 5

    =

    = =

    13) 2 x

    x 2 =

    2x x

    2

    2

    +

    +

    x2 (x2 + 2) = (2 x) (2 + x) x4 + 2x2 = 4 x2 x4 + 3x2 4 = 0 y2 + 3y 4 = 0

    2 1 1

    2

    3 5y 1 x 1

    3 ( 3) 41( 4) 3 5 2y3 521 2

    y 42

    + = = = = = = =

    14) 2

    22

    3x 2 = 4x 3

    x 2

    + ++

    3x2 + 2 = (4x2 + 3)(x2 + 2) 3x2 + 2 = 4x4 + 8x2 + 3x2 + 6 4x4 + 8x2 + 4 = 0 y2 + 2y + 1 = 0 (y + 1)2 = 0

    y = -1 x2 = -1 No hay solucin real

    15) 1

    14

    =+

    x

    x

    x

    x

    (x + 4)(x 1) x(x 1) = x2 x2 + 4x x 4 x2 + x = x2 x2 4x + 4 = 0 (x 2)2 = 0 x 2=

    16) 2

    141

    24

    =+

    xx

    24(x 2) + x(x 2) = 14x 24x 48 + x2 2x = 14x x2 + 8x 48 = 0

    2 1 1

    2 2

    8 16x 4 x 4

    8 ( 8) 41( 48) 8 16 2x8 1621 2

    x 12 x 122

    + = = = = = = = =

    17) 21

    1

    4 =++ x

    x

    x

    4x + (x + 1)(x 1) = 2x(x + 1) 4x + x2 1 = 2x2 + 2x x2 2x + 1 = 0 (x 1)2 = 0 x 1=

    18) x

    x

    x

    x

    2

    23

    2

    2 =+

    4x2 = (3x 2)(x + 2) 4x2 = 3x2 + 6x 2x 4 x2 4x + 4 = 0 (x 2)2 = 0 x 2=

  • ECUACIONES 1 BCT

    Luisa Muoz 4

    19) 2

    6

    2

    8

    +=+

    xx

    xx

    x 6 8x

    x 2 x 2

    +=

    x 2

    xx 2

    =

    x 1=

    20) 21

    2 =

    x

    x

    x

    x

    (x 2)(x 1) x2 = -2x(x 1) x2 2x x + 2 x2 = - 2x2 + 2x 2x2 5x + 2 = 0

    1 12

    2 2

    5 3x 2 x 2

    45 5 422 5 3x

    5 3 1 122 4x x

    4 2 2

    + = = = = = = = =

    21) 5

    1323

    1

    5 =+

    x

    x

    x

    x

    5x(x 5) + (5x 5)(3x 2) = 13x(x 1) 5x2 25x + 15x2 10x 15x + 10 = 13x2 13x 7x2 37x + 10 = 0

    1 12

    2 2

    37 33x 5 x 5

    1437 37 4710 37 33x

    37 33 4 227 14x x

    14 14 7

    + = = = = = = = =

    22) 6

    131

    1

    4 =++

    x

    x

    x

    x

    6x(x + 4) 6(x 1)(x + 1) = 13x(x 1) 6x2 + 24x 6x2 + 6 = 13x2 13x 13x2 37x 6 = 0

    1 12

    2 2

    37 41x 3 x 3

    2637 37 413( 6) 37 41x

    37 41 4 2213 26x x

    26 26 13

    + = = = = = = = =

    23) 5

    224

    5

    105

    2

    ++=

    +

    x

    xx

    xx

    24x 22x 105x

    x 5

    + +=+

    5x2 + 25x = 4x2 + 22x + 10 x2 + 3x 10 = 0

    2 1 1

    2 2

    3 7x 2 x 2

    3 ( 3) 4( 10) 3 7 2x3 72 2

    x 5 x 52

    += = = = = = = =

  • ECUACIONES 1 BCT

    Luisa Muoz 5

    24) 6

    13

    2x

    1x

    1x

    2x =+++

    ++

    6(x + 1)2 + 6(x + 2)2 = 13(x + 1)(x + 2) 6x2 + 12x + 6 + 6x2 + 24x + 24 = 13x2 + 26x + 13x + 26

    x2 + 3x 4 = 0 2 1 1

    2 2

    3 5x 1 x 1

    3 ( 3) 4( 4) 3 5 2x3 52 2

    x 4 x 42

    + = = = = = = = =

    25) 2x2

    2x

    2x

    x2=

    ++

    +

    4x2 + (x + 2)2 = 4x (x + 2) 4x2 + x2 + 4x + 4 = 4x2 + 8x x2 4x + 4 = 0 (x 2)2 = 0 x 2=

    26) 14x

    x

    1x

    x=

    ++

    +

    x(x + 4) + x(x + 1) = (x + 4)(x + 1) x2 + 4x + x2 + x = x2 + 4x + x + 4 x2 = 4 x 2=

    27) 2

    9

    2x

    2

    1x

    1=

    ++

    +

    2(x + 2) + 4(x + 1) = 9(x + 1)(x + 2) 2x + 4 + 4x + 4 = 9x2 + 18x + 9x + 18 9x2 + 21x + 10 = 0

    1 12

    2 2

    21 9 12 2x x

    18 18 321 ( 21) 4910 21 9x

    29 18 21 9 30 5x x

    18 18 3

    += = = = = = = =

    28) 2x

    2

    6

    1

    3x

    3

    =+

    +

    18(x 2) + (x 2)(x + 3) = 12(x + 3) 18x 36 + x2 + 3x 2x 6 = 12x + 36 x2 + 7x 78 = 0

    2 1 1

    2 2

    7 19x 6 x 6

    7 ( 7) 4( 78) 7 19 2x7 192 2

    x 13 x 132

    + = = = = = = = =

    29) 2x

    9x

    1x

    7x

    1x

    1x2

    ++

    =

    +

    (2x 1)(x2 + x 2) (x 7)(x2 + 3x + 2) = (x + 9)(x2 1) 2x3 + 2x2 4x x2 x + 2 x3 3x2 2x + 7x2 +

    21x + 14x = x3 x + 9x2 9 4x2 15x 25 = 0

    1 12

    2 2

    15 25x 5 x 5

    815 15 44( 25) 15 25x

    15 25 10 524 8x x

    8 8 4

    + = = = = = = = =

  • ECUACIONES 1 BCT

    Luisa Muoz 6

    30) 6

    13

    x

    1x

    1x

    x=

    ++

    +

    6x2 + 6(x + 1)2 = 13x(x + 1) 6x2 + 6x2 + 12 x + 6 = 13x2 + 13x x2 + x 6 = 0

    2 1 1

    2 2

    1 5x 2 x 2

    1 ( 1) 4( 6) 1 5 2x1 52 2

    x 3 x 32

    + = = = = = = = =

    31) 3x2x

    12

    1x

    3

    3x

    x2 +

    =

    ++

    x(x 1) + 3(x + 3) = 12 x2 x + 3x + 9 = 12 x2 + 2x 3 = 0

    2 1 1

    2 4x 1 x 1

    2 ( 2) 4( 3) 2 4 2x2 2

    += = = = =

    2 2

    2 4x 3 x 3

    2

    = = =

    No hay solucin

    32) 2

    1 2x1

    x 1 1 x+ =

    +

    1 x + 2x = 1 x2 1 x + 2x = 1 x2 x2 + x = 0 x(x + 1) = 0 1

    2

    x 0

    x 1 0 x 1

    =

    + = =

    33) 16

    24

    4

    4

    4

    42

    =+

    +

    xx

    x

    x

    x

    (x + 4)2 (x 4)2 = 24 x2 +8x + 16 x2 + 8x 16 = 24 16x = 24 3

    x2

    =

    34) 32

    20

    1

    5

    3 2 =

    +

    xxxxx

    x(x + 1) 5(x 3) = 20 x2 + x 5x + 15 = 20 x2 4x 5 = 0

    2 1 1

    2 2

    4 6x 5 x 5

    4 4 4( 5) 4 6 2x4 62 2

    x 1 x 12

    += = = = = = = =

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