how to solve complex quadratic?

Question

Can someone tell me the steps to solve this?
x^2=-i
the answer is +-(1-i)/sqrt(2)
but I have no idea how to arrive at that answer and know I'm just missing something silly. Thanks!

Answer 1

wait...
any complex number in a+bi form (rectangular) can be rewritten in r∙cisΘ form (polar).
Note:
a²+b²=r²
r∙cisΘ = r∙cosΘ+i∙r∙sinΘ = x + i∙y. You need trigonometry.

Now to get the solutions to xⁿ=tⁿcisΘ.
We have x = t cis([Θ+360°k]/n) , k runs from 0 to n-1.

Now, x² = -i=cis270° (r = 1)

x1=cis135° and x2=cis(135°+180°)=cis315°
I believe that you can obtain the rectangular form of the two answers using trigonometry.

Answer 2

Try expressing x as a complex number, whose real & imaginary parts you don't yet know (call them a and b):

x = a + bi

Now the equation becomes:

(a + bi)² = -i
a² + 2abi + b²i² = -i
a² + 2abi + b²(-1) = -i
(a² – b²) + 2abi = -i

Now, the "real part" of the left side must equal the "real part" of the right side. Thus:

a² – b² = 0

Likewise, the "imaginary part" of the left side must equal the "imaginary part" of the right side. Thus:

2ab = -1

You can then solve these two equations simultaneously to get a and b. That in turn gives you x.