How do you find complex roots? Is there an equation?

Question

For example, I used Maple to find the roots of:
(r^4 + 5r^2 +4=0). How do you find the roots by hand? thanks

Answer 1

Note that all powers of r in the given equation are even.

Let x = r^2. This will reduce he equation to a quadratic equation which you can solve by factorization.

x^2 + 5x + 4 = 0
(x+4)(x+1)=0
x={-4, -1}

r = ±sqrt(x)
r = {-2i, -i, i, 2i}

Answer 2

Do the following:
Consider that r^2=x.
Your ecuation will now be x^2 +5x +4 =0, with the roots -1 and -4, if i'm not mistaken.
Now, reverting to your previous notations, you have r^2 = -1 and r^2=-4.
r^2 = -1 leads to r = +i and -i
r^2 = -4 leads to r=sqrt(-4)= sqrt(-1) x sqrt(4)= +2i or -2i.
I hope this clears things up. Feel free to contact me for more questions

Answer 3

r ² = [ - 5 ± √(25 - 16) ] / 2
r ² = [ - 5 ± 3 ] / 2
r ² = - 1 , r ² = - 4
r ² = i ² , r² = 4 i ²
r = ± i , r = ± 2i

Answer 4

guess and check theres no other way to find complex roots by hand unless you know the exponent opposite

Answer 5

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