Find all distinct roots (real or complex) of *z2*+(−1+*i*)*z*+(6−18*i*). Enter the roots as a comma-separated list of values of the form *a*+*bi*.

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.

Find all distinct roots (real or complex) of z^2 + (−1 + i)z + (6 − 18i). Enter the roots as a comma-separated list of values of the form a + bi.

Explore the process of solving the complex quadratic equation z^2 + (-1 + i)z + (6 - 18i). This detailed step-by-step solution covers complex number operations, the quadratic formula, and the square root of complex numbers. Suitable for advanced high school students in Grades 11-12.

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