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*. Use the square root symbol '√' where needed to give an exact value for your answer.

The given quadratic equation is:

\[
z^2 + (-1 + i)z + (6 - 18i) = 0
\]

This is a quadratic equation in 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: Compute \( b^2 \)
\[
b = -1 + i
\]
\[
b^2 = (-1 + i)^2 = (-1)^2 + 2(-1)(i) + i^2 = 1 - 2i - 1 = -2i
\]
So, \( b^2 = -2i \).

### Step 2: Compute \( 4ac \)
\[
a = 1, \quad c = 6 - 18i
\]
\[
4ac = 4 \times 1 \times (6 - 18i) = 24 - 72i
\]

### Step 3: Compute the discriminant \( b^2 - 4ac \)
\[
b^2 - 4ac = -2i - (24 - 72i) = -2i - 24 + 72i = -24 + 70i
\]

### Step 4: Compute the square root of the discriminant
We need to find \( \sqrt{-24 + 70i} \). To do this, express \( -24 + 70i \) in polar form and then take the square root.

The magnitude \( r \) of \( -24 + 70i \) is:
\[
r = \sqrt{(-24)^2 + (70)^2} = \sqrt{576 + 4900} = \sqrt{5476} = 74
\]

The argument \( \theta \) is:
\[
\theta = \text{atan2}(70, -24) \approx 108.37^\circ \text{ or } 1.891 \, \text{radians}
\]

Taking the square root in polar form:
\[
\sqrt{74} \left( \cos \frac{1.891}{2} + i \sin \frac{1.891}{2} \right)
\]
\[
= \sqrt{74} \left( \cos 0.9455 + i \sin 0.9455 \right)
\]
\[
= \sqrt{74} (0.584 + i 0.812)
\]
\[
\approx 8.6 + 11.2i
\]

Thus, \( \sqrt{-24 + 70i} \approx 8.6 + 11.2i \).

### Step 5: Apply the quadratic formula
Now substitute back into the quadratic formula:

\[
z = \frac{-(-1 + i) \pm (8.6 + 11.2i)}{2}
\]
\[
z = \frac{1 - i \pm (8.6 + 11.2i)}{2}
\]

This gives two solutions:

1. \( z_1 = \frac{1 - i + (8.6 + 11.2i)}{2} = \frac{9.6 + 10.2i}{2} = 4.8 + 5.1i \)
2. \( z_2 = \frac{1 - i - (8.6 + 11.2i)}{2} = \frac{-7.6 - 12.2i}{2} = -3.8 - 6.1i \)

### Final Answer:
The distinct roots of the equation are:

\[
z_1 = 4.8 + 5.1i, \quad z_2 = -3.8 - 6.1i
\]

Would you like any clarifications or further details?

### Related Questions:
1. How do you solve complex quadratic equations in polar form?
2. What is the significance of the discriminant in complex roots?
3. How can you simplify square roots of complex numbers?
4. Can the quadratic formula always be used with complex coefficients?
5. What are some real-world applications of solving complex quadratic equations?

### Tip:
When working with complex numbers, converting them to polar form often simplifies operations like finding square roots.

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. Use the square root symbol '√' where needed to give an exact value for your answer.

Discover how to find the distinct complex roots of the quadratic equation z^2 + (-1 + i)z + (6 - 18i) = 0. This problem involves solving quadratic equations with complex numbers, utilizing the quadratic formula, and calculating square roots in polar form. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation