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

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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.

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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.

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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

(x^2 + 4) (x^2 - 16)