Find an expression for the attenuation from the eqn given

[BadPhysicistAtWork]

Having a big issue working through this problem and was wondering if somebody, much smarter than I, could give me a few pointers. Anyway here's the problem:

Given that k²=εμω²-iωμσ

By writing the wave number as k=k_r-ik_i.
Show that k_i, which determines attenuation, can be expressed by:
k_i=ω(εμ/2)^1/2[(1+(σ²/ω²ε²))^1/2)-1]^1/2

I've started by equating real and imaginary parts however I find an impasse where I have a quartic equation.

I'd been incredibly grateful for any answers, thank you in advance!

Just as a quick note, I followed the following process but reached an impasse (may be completely off base here):
k²=εμω²-iωμσ
Substitute k=k_r-ik_i. into the equation which gives;
k_r²-2ik_rk_i-k_i²=εμω²-iωμσ
Then split into two equations by equating real and imaginary parts;
(1) k_r²-k_i²=εμω²
(2) εμω²=ωμσ/2k_i
Now (2) into (1) gives;
(ωμσ/2k_i)²-k_i²=εμω²
Then rearranging gives;
k_i⁴+εμω²k_i²=ω²μ²σ²/4

[EDIT: Realised i'd been a bit of an idiot and missed what should have been quite obvious, thank you for pointing it out!]

 

[Dr Transport]

it is a quadratic... set k_{i} = \alpha and us the quadratic equation for \alpha and take the square root...

 

[BadPhysicistAtWork]

Actually really disappointed i myself for not seeing that... Thank you for that, really made this a lot easier.