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30 March, 22:21

Solve x^4-12x^2-64=0

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Answers (2)
  1. 30 March, 22:48
    0
    x

    4

    -

    12

    x

    2

    =

    64

    x

    4

    -

    12

    x

    2

    =

    64

    Move

    64

    64

    to the left side of the equation by subtracting it from both sides.

    x

    4

    -

    12

    x

    2

    -

    64

    =

    0

    x

    4

    -

    12

    x

    2

    -

    64

    =

    0

    Rewrite

    x

    4

    x

    4

    as

    (

    x

    2

    )

    2

    (

    x

    2

    )

    2

    .

    (

    x

    2

    )

    2

    -

    12

    x

    2

    -

    64

    =

    0

    (

    x

    2

    )

    2

    -

    12

    x

    2

    -

    64

    =

    0

    Let

    u

    =

    x

    2

    u

    =

    x

    2

    . Substitute

    u

    u

    for all occurrences of

    x

    2

    x

    2

    .

    u

    2

    -

    12

    u

    -

    64

    =

    0

    u

    2

    -

    12

    u

    -

    64

    =

    0

    Factor

    u

    2

    -

    12

    u

    -

    64

    u

    2

    -

    12

    u

    -

    64

    using the AC method.

    Tap for fewer steps ...

    Consider the form

    x

    2

    +

    b

    x

    +

    c

    x

    2

    +

    b

    x

    +

    c

    . Find a pair of integers whose product is

    c

    c

    and whose sum is

    b

    b

    . In this case, whose product is

    -

    64

    -

    64

    and whose sum is

    -

    12

    -

    12

    .

    -

    16

    ,

    4

    -

    16

    ,

    4

    Write the factored form using these integers.

    (

    u

    -

    16

    )

    (

    u

    +

    4

    )

    =

    0

    (

    u

    -

    16

    )

    (

    u

    +

    4

    )

    =

    0

    Replace all occurrences of

    u

    u

    with

    x

    2

    x

    2

    .

    (

    x

    2

    -

    16

    )

    (

    x

    2

    +

    4

    )

    =

    0

    (

    x

    2

    -

    16

    )

    (

    x

    2

    +

    4

    )

    =

    0

    Rewrite

    16

    16

    as

    4

    2

    4

    2

    .

    (

    x

    2

    -

    4

    2

    )

    (

    x

    2

    +

    4

    )

    =

    0

    (

    x

    2

    -

    4

    2

    )

    (

    x

    2

    +

    4

    )

    =

    0

    Since both terms are perfect squares, factor using the difference of squares formula,

    a

    2

    -

    b

    2

    =

    (

    a

    +

    b

    )

    (

    a

    -

    b

    )

    a

    2

    -

    b

    2

    =

    (

    a

    +

    b

    )

    (

    a

    -

    b

    )

    where

    a

    =

    x

    a

    =

    x

    and

    b

    =

    4

    b

    =

    4

    .

    (

    x

    +

    4

    )

    (

    x

    -

    4

    )

    (

    x

    2

    +

    4

    )

    =

    0

    (

    x

    +

    4

    )

    (

    x

    -

    4

    )

    (

    x

    2

    +

    4

    )

    =

    0

    If any individual factor on the left side of the equation is equal to

    0

    0

    , the entire expression will be equal to

    0

    0

    .

    x

    +

    4

    =

    0

    x

    +

    4

    =

    0

    x

    -

    4

    =

    0

    x

    -

    4

    =

    0

    x

    2

    +

    4

    =

    0

    x

    2

    +

    4

    =

    0

    Set the first factor equal to

    0

    0

    and solve.

    Tap for fewer steps ...

    Set the first factor equal to

    0

    0

    .

    x

    +

    4

    =

    0

    x

    +

    4

    =

    0

    Subtract

    4

    4

    from both sides of the equation.

    x

    =

    -

    4

    x

    =

    -

    4

    Set the next factor equal to

    0

    0

    and solve.

    Tap for more steps ...

    x

    =

    4

    x

    =

    4

    Set the next factor equal to

    0

    0

    and solve.

    Tap for more steps ...

    x

    =

    2

    i

    ,

    -

    2

    i

    x

    =

    2

    i

    ,

    -

    2

    i

    The final solution is all the values that make

    (

    x

    +

    4

    )

    (

    x

    -

    4

    )

    (

    x

    2

    +

    4

    )

    =

    0

    (

    x

    +

    4

    )

    (

    x

    -

    4

    )

    (

    x

    2

    +

    4

    )

    =

    0

    true.

    x

    =

    -

    4

    ,

    4

    ,

    2

    i

    ,

    -

    2

    i

    x

    =

    -

    4

    ,

    4

    ,

    2

    i

    ,

    -

    2

    i

    x

    4

    -

    1

    2

    x

    2

    =

    6

    4

    x

    4

    -

    1

    2

    x

    2

    =

    6

    4
  2. 31 March, 00:06
    0
    (x^2 + 4) (x^2 - 16)
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