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The given quadratic equation is:

\[
z^2 - (3 + 5i)z + 8i - 5 = 0
\]

We will solve this using the quadratic formula, which is:

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

For the given quadratic equation, the coefficients are:
- \( a = 1 \) (coefficient of \( z^2 \)),
- \( b = -(3 + 5i) \) (coefficient of \( z \)),
- \( c = 8i - 5 \) (constant term).

### Step 1: Simplify the values of \( b \) and \( c \)
- \( b = -(3 + 5i) = -3 - 5i \),
- \( c = 8i - 5 \).

### Step 2: Substitute into the quadratic formula

Now, apply these values to the quadratic formula:

\[
z = \frac{-(-3 - 5i) \pm \sqrt{(-3 - 5i)^2 - 4(1)(8i - 5)}}{2(1)}
\]

Simplifying the numerator:

\[
z = \frac{3 + 5i \pm \sqrt{(-3 - 5i)^2 - 4(8i - 5)}}{2}
\]

### Step 3: Compute the discriminant

Now, calculate the discriminant \( \Delta = b^2 - 4ac \).

#### 3.1: Compute \( (-3 - 5i)^2 \)
\[
(-3 - 5i)^2 = (-3)^2 + 2(-3)(-5i) + (-5i)^2 = 9 + 30i + 25i^2 = 9 + 30i - 25 = -16 + 30i
\]

#### 3.2: Compute \( 4ac = 4(1)(8i - 5) = 4(8i - 5) = 32i - 20 \)

#### 3.3: Calculate the discriminant \( \Delta \)

\[
\Delta = (-16 + 30i) - (32i - 20) = -16 + 30i - 32i + 20 = 4 - 2i
\]

### Step 4: Solve for \( z \)

Now, we have the discriminant \( \Delta = 4 - 2i \). Taking the square root of \( 4 - 2i \) requires converting it to polar form.

#### 4.1: Convert \( 4 - 2i \) to polar form

The magnitude \( r \) is:

\[
r = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
\]

The argument \( \theta \) is:

\[
\theta = \text{arg}(4 - 2i) = \tan^{-1}\left(\frac{-2}{4}\right) = \tan^{-1}\left(-\frac{1}{2}\right)
\]

Thus, the polar form of \( 4 - 2i \) is \( 2\sqrt{5} \left(\cos(\theta) + i \sin(\theta)\right) \), but to keep things simple, we'll use an approximate form of \( \sqrt{4 - 2i} \), which will give us two possible square roots.

#### 4.2: Approximate \( \sqrt{4 - 2i} \)

The square root of \( 4 - 2i \) can be approximated as:

\[
\sqrt{4 - 2i} \approx 2.23 - 0.45i
\]

### Step 5: Substitute and solve

Now, substitute the square root value into the quadratic formula:

\[
z = \frac{3 + 5i \pm (2.23 - 0.45i)}{2}
\]

This gives two possible solutions:

1. \( z_1 = \frac{3 + 5i + (2.23 - 0.45i)}{2} = \frac{5.23 + 4.55i}{2} = 2.615 + 2.275i \)

2. \( z_2 = \frac{3 + 5i - (2.23 - 0.45i)}{2} = \frac{0.77 + 5.45i}{2} = 0.385 + 2.725i \)

### Final Solutions:

\[
z_1 = 2.615 + 2.275i
\]
\[
z_2 = 0.385 + 2.725i
\]

These are the solutions to the quadratic equation.

---

Would you like more details on any part of the solution?

### Related Questions:
1. How do you solve a quadratic equation with complex coefficients?
2. What is the polar form of a complex number, and how do you convert to it?
3. How do you compute the square root of a complex number?
4. Can all quadratic equations have complex solutions?
5. What is the significance of the discriminant in complex number equations?

### Tip:
When solving quadratic equations with complex numbers, it can be helpful to use polar coordinates to simplify operations like taking square roots.

Solve the quadratic equation z^2 - (3 + 5i)z + 8i - 5 = 0

Learn how to solve the quadratic equation z^2 - (3 + 5i)z + 8i - 5 = 0, which involves complex numbers. This solution covers the quadratic formula and the method for finding square roots of complex numbers using polar form. Suitable for advanced students at the undergraduate level.

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