Complex numbers

[calcrik]

Hello.

I have been stuck on these two problems forever.

Find all four roots of the equation:

x⁴-2x²+4 = 0

As far as I know the only way to find an answer here is by factoring, but I can't seem to figure it out.
I don't know all the rules when factoring with complex numbers.

One of my failed attempts looked like this: (x²-1+i)(x²+1+i), (didn't go further).


The other problem:

Evaluate the limit:

lim |x-4|-2
x->2 (x-2)

Does this not exist? I tried multiplying with (x+2), but it didn't get me anywhere.

I also tried approaching 2 from both sides. I remember an example in my calculus book where the denominator looked like this(absolute value in numerator): (x-2)(x+3). When approaching 2 from left, or right, this didn't result in 0(2+3), but rather a positive or negative outcome. Can I do the same here? And if not, why?

Any help would be appreciated

 

[burakaltr]

Hello.

I have been stuck on these two problems forever.

Find all four roots of the equation:

x⁴-2x²+4 = 0

As far as I know the only way to find an answer here is by factoring, but I can't seem to figure it out.
I don't know all the rules when factoring with complex numbers.

One of my failed attempts looked like this: (x²-1+i)(x²+1+i), (didn't go further).


The other problem:

Evaluate the limit:

lim |x-4|-2
x->2 (x-2)

Does this not exist? I tried multiplying with (x+2), but it didn't get me anywhere.

I also tried approaching 2 from both sides. I remember an example in my calculus book where the denominator looked like this(absolute value in numerator): (x-2)(x+3). When approaching 2 from left, or right, this didn't result in 0(2+3), but rather a positive or negative outcome. Can I do the same here? And if not, why?

Any help would be appreciated

x⁴-2x²+4 = 0

( x**(2) - 1) ** 2 + 3 =0

((x**2) -1 )**2 =-3

take roots of both sides to get, but + then -

((x**2) -1 ) = sqrt(3) i

or

((x**2)-1 )= -sqrt(3) i

From these you should be able to get 4 roots

 

[burakaltr]

lim |x-4|-2
x->2 (x-2)

upper side would negate as x-4<0 when x=2

take it out as

4-x

then

( 4-x-2 )/ ( x- 2) = you tell us

 

[Deleted member 4993]

You should plot the function to see "what you should get" as the answer.