[Math] Finding values for K such that the roots of the quadratic are strictly imaginary: $x^2+\left(K^2+3K-7\right)x+K$

Question

$x^2+\left(K^2+3K-7\right)x+K$

I'd like to know the general approach needed to find out how to find solutions for K when I want the roots of this equation to have a specific property, such as strictly imaginary, or when I want them to have a positive real part only, or negative real part only.

Bonus points if the approach applies to a general quadratic of the form $ax^2+bx+c$, what equations should be satisfied if I want specific properties for the roots? Extra bonus points for the same question, but for cubic and quartic equations.

Answer 1

Hint: The roots of a monic quadratic are strictly imaginary iff it is of the form $(x-bi)(x+bi)=x^2+b^2$.