We are tasked with finding the distinct roots of the quadratic equation:

$z^2 + (-1 + i)z + (6 - 18i) = 0$

This is a quadratic equation of the form $az^2 + bz + c = 0$, where:

• $a = 1$,
• $b = -1 + i$,
• $c = 6 - 18i$.

To find the roots, we use the quadratic formula:

$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step 1: Identify the coefficients

The coefficients are:

• $a = 1$,
• $b = -1 + i$,
• $c = 6 - 18i$.

Step 2: Compute the discriminant

The discriminant is given by:

$\Delta = b^2 - 4ac$

Substitute $b = -1 + i$, $a = 1$, and $c = 6 - 18i$:

$\Delta = (-1 + i)^2 - 4(1)(6 - 18i)$

First, calculate $(-1 + i)^2$:

$(-1 + i)^2 = (-1)^2 + 2(-1)(i) + i^2 = 1 - 2i - 1 = -2i$

Next, calculate $4ac$:

$4ac = 4(6 - 18i) = 24 - 72i$

Now, compute the discriminant:

$\Delta = -2i - (24 - 72i) = -2i - 24 + 72i = 70i - 24$

Thus, the discriminant is:

$\Delta = -24 + 70i$

Step 3: Compute the square root of the discriminant

We need to find the square root of $\Delta = -24 + 70i$. To do this, we express it in polar form and then take the square root.

• The modulus $r$ is given by:

$r = \sqrt{(-24)^2 + (70)^2} = \sqrt{576 + 4900} = \sqrt{5476} = 74$

• The argument $\theta$ is:

$\theta = \tan^{-1}\left(\frac{70}{-24}\right) = \tan^{-1}\left(\frac{-70}{24}\right) \approx 1.854 \text{ radians}$

The square root of $\Delta = 74 \text{cis}(1.854)$ is:

[ \sqrt{\Delta} = \sqrt{74} \text{cis}\left(\frac{1.