Solving a quadratic equation with imaginary numbers in it

So I don’t get why when you are solving a quadratic equation that has imaginary numbers in it, you have to separate the imaginary parts and the real parts to find a solution.

Let’s take for example this question. Take note that “i” stands for imaginary number.

If 1+ki is a solution of 2z^2 -kz+34=0, find the value of k.

So we throw in the 1+ki and expand them. Then we get this formula:
2 + 4ki - 2k^2 -k - ik^2 + 34

And that is where you separate the real parts and imaginary parts then put the real parts to 0 to solve.
Real: 2k^2 - k + 36 = 0
Why do we do that?

Welcome to the forum @icymonkey2004 !

We know that two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal . What you do when you separate these parts is you apply this principle, or what follows from this principle - to equal zero, a complex number must have both its real part and its imaginary part equal to zero.

2 + 4ki - 2k^2 -k - ik^2 + 34 = 0

2k^2 + k - 36 = 0

Then you solve the equation.

Does it answer your question?