How to solve 2nd degree complex polynomials with complex numbers as coefficients?

Question

I tried to solve this complex polynomial:

$$(1 + i)z^2 + (-1 + 7i)z - (10 - 2i) = 0$$

When I did the whole discriminant thing and I got $\sqrt{18i}$ which I don't know how to deal with. Usually my dicriminant doesn't contain the i inside of it. How are equations like these solved?

Thank you.

Answer 1

You have to look for the square roots of $18i$ In the complex plane. The most common way of doing so is to put your complex number in polar form, here $18i=18 e^{i\pi/2}$. Then look for square roots of $18$ (should be easy) and $e^{i\pi/2}$.

Answer 2

Hint: Notice that $$18i = 9\cdot 2i = 9 \cdot (1+i)^2$$

Answer 3

There is nothing at all wrong with representing the solutions by

$$x = \frac{(1 - 7i) + \sqrt{18i}}{2(1 + i)}$$

That is a perfectly valid, explicit mathematical expression on the right and if $x$ satisfies this equation, it satisfies the other. The trick is that, as mentioned in the comments, $\sqrt{a}$ is ambiguous for complex $a$-input - technically it is too for reals, but "less" so in that it is easier to choose between the two options. But as long as you understand that symbol as being substitutible for either of two values it represents, or you set up a convention ahead of time, then there is no problem here.