SOLUTION: Find the value of k in x^4-4x^2+k that gives 4 complex zeros.

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Question 1102632: Find the value of k in x^4-4x^2+k that gives 4 complex zeros.
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
in order for the equation to give you 4 complex zeros, the equaiton should not cross or touch the x-axis.

set the equation equal to 0 to get:

x^4 - 4x^2 + k = 0

let y = x^2.

the equation becomes y^2 - 4y + k = 0

this is a quadratic equation that can be solved through various means.

i'm not sure if you have to do it this way, but the use of the quadratic formula should be helpful.

in fact, if the discriminant is negative, then you will have complex zeroes.

now that you have the formula in standard form of ay^2 + by + c = 0, you can use the quadratic formula as shown below:

your formula is y^2 - 4y + k = 0
a = 1
b = -4
c = k
use of the quadratic formula will get you:

y = -b + sqrt(b^2 - 4ac) / (2a) or y = -b - sqrt(b^2 - 4ac) / (2a)

b^2 - 4ac is the discriminant.

when b = -4 and a = 1 and c = k, you get:

b^2 - 4ax = (-4)^2 - 4*1*k)

this simplifies to 16 - 4k

is 16 - 4k is negative, then the solution will be complex.

set 16 - 4k < 0

add 4k to both sides of this equation to get 16 < 4k

divide both sides of this equation by 4 to get 4 < k

this means that k > 4

any value of k > 4 should do the trick.

we'll use 5 as a test.

when k = 5, your equation becomes y^2 - 4y + 5 = 0

solve this using quadratic formula to get:

y = 2+i or 2-i

since you have previously set y = x^2, then you get:

x^2 = 2+i or x^2 = 2-i

this leads to x = plus or minus sqrt(2+i) or x = plus or minus sqrt(2-i).

that's 4 complex roots of the equation x^4 - 4x^2 + 5 = 0

your factors will be (x-sqrt(2+i)) * (x+sqrt(2+i)) * (x-sqrt(2-i)) * (x+sqrt(2-i))

if you multiply those factors together, you will get x^4 - 4x^2 + 5.

if you graph x^4 - 4x^2 + 5, you will see that it doesn't cross the x-axis.

since it is a degree 4 equation, then it must have 4 roots, therefore all the roots have to be complex.

here's the graph.

$$$



Answer by ikleyn(52805)   (Show Source): You can put this solution on YOUR website!
.
The necessary and sufficient condition for it is THIS:


    The polynomial (the function)  x^4 - 4x^2 + k  is always positive (and never zero or negative).


In turn, the necessary and sufficient condition for it is THIS:

   
    The polynomial  y^2 -4y + k  is always positive.


In turn, the necessary and sufficient condition for it is THIS:


    The discriminant  d = 4^2 - 4k is negative:


    16 - 4k < 0  ====>  16 < 4k  ====>  4 < k.


Answer.  k > 4.


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