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.