To find the distinct roots of the given quadratic equation in the form:

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

We'll use the quadratic formula, which is given by:

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

For this equation, we have:

• $a = 1$ (the coefficient of $z^2$),
• $b = -1 + i$ (the coefficient of $z$),
• $c = 6 - 18i$ (the constant term).

Step 1: Identify $a$, $b$, and $c$

$a = 1, \quad b = -1 + i, \quad c = 6 - 18i$

Step 2: Compute the discriminant, $\Delta = b^2 - 4ac$

We need to compute $b^2$ and $4ac$ separately.

(i) Calculate $b^2$:

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

(ii) Calculate $4ac$:

$4ac = 4 \times 1 \times (6 - 18i) = 4(6 - 18i) = 24 - 72i$

(iii) Compute the discriminant:

$\Delta = b^2 - 4ac = -2i - (24 - 72i) = -2i - 24 + 72i = 70i - 24$ Thus, the discriminant is: $\Delta = -24 + 70i$

Step 3: Compute $\sqrt{\Delta}$

To calculate $\sqrt{-24 + 70i}$, we first convert it to polar form.

• The modulus $r$ is given by $r = \sqrt{(-24)^2 + (70)^2} = \sqrt{576 + 4900} = \sqrt{5476} = 74$.
• The argument $\theta$ is $\tan^{-1}\left( \frac{70}{-24} \right) = \tan^{-1}\left( -\frac{70}{24} \right) \approx 108.43^\circ$.

Thus, in polar form: $\sqrt{-24 + 70i} = \sqrt{74} \left( \cos\left( \frac{\theta}{2} \right) + i \sin\left( \frac{\theta}{2} \right) \right)$

Calculating further will give the square root.