We are tasked with solving the quadratic equation:

$z^2 + (6 - 4i)z + 10 = 0$

To solve this, we will use the quadratic formula:

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

For the given equation, the coefficients are:

• $a = 1$ (coefficient of $z^2$),
• $b = 6 - 4i$ (coefficient of $z$),
• $c = 10$ (constant term).

Substituting these values into the quadratic formula:

$z = \frac{-(6 - 4i) \pm \sqrt{(6 - 4i)^2 - 4(1)(10)}}{2(1)}$

First, let's compute the discriminant ($b^2 - 4ac$):

$(6 - 4i)^2 = 6^2 - 2(6)(4i) + (4i)^2 = 36 - 48i - 16 = 20 - 48i$ $4ac = 4(1)(10) = 40$ So the discriminant is:

$20 - 48i - 40 = -20 - 48i$

Now, we substitute the discriminant back into the quadratic formula:

$z = \frac{-(6 - 4i) \pm \sqrt{-20 - 48i}}{2}$

The next step involves calculating $\sqrt{-20 - 48i}$. To do this, we will express the complex number in polar form and then take the square root.

1. First, find the modulus $r$ of $-20 - 48i$:

$r = \sqrt{(-20)^2 + (-48)^2} = \sqrt{400 + 2304} = \sqrt{2704} = 52$

2. Next, find the argument $\theta$ (the angle):

$\theta = \tan^{-1}\left(\frac{-48}{-20}\right) = \tan^{-1}(2.4)$

Approximating $\tan^{-1}(2.4)$, we get $\theta \approx 1.176$ radians.

3. Now we take the square root in polar form. The square root of a complex number $r(\cos \theta + i \sin \theta)$ is:

$\sqrt{r} \left( \cos\left(\frac{\theta}{2}\right) + i \sin\left(\frac{\theta}{2}\right) \right)$

Thus:

$\sqrt{52} \left( \cos\left(\frac{1.176}{2}\right) + i \sin\left(\frac{1.176}{2}\right) \right) = 7.211 \left( \cos(0.588) + i \sin(0.588) \right)$

Approximating $\cos(0.588) \approx 0.832$ and $\sin(0.588) \approx 0.554$, we get:

$\sqrt{-20 - 48i} \approx 7.211(0.832 + 0.554i) = 6.002 + 4.0i$

Now, we substitute this back into the quadratic formula:

$z = \frac{-(6 - 4i) \pm (6.002 + 4.0i)}{2}$

This gives us two solutions:

1. For the plus sign:

$z = \frac{-(6 - 4i) + (6.002 + 4.0i)}{2} = \frac{-6 + 4i + 6.002 + 4.0i}{2} = \frac{0.002 + 8i}{2} = 0.001 + 4i$

2. For the minus sign:

$z = \frac{-(6 - 4i) - (6.002 + 4.0i)}{2} = \frac{-6 + 4i - 6.002 - 4.0i}{2} = \frac{-12.002 + 0i}{2} = -6.001$

Thus, the solutions to the quadratic equation are:

$z_1 = 0.001 + 4i \quad \text{and} \quad z_2 = -6.001$

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Would you like a more detailed explanation of any of the steps? Here are some related questions that may be helpful:

1. How do you find the modulus and argument of a complex number?
2. What is the geometric interpretation of the square root of a complex number?
3. How can you verify the solutions to a quadratic equation using substitution?
4. How do you solve a quadratic equation when the coefficients are complex numbers?
5. What is the significance of the discriminant in a quadratic equation?

Tip: When dealing with complex numbers, always express them in polar form before performing operations like taking square roots or raising to powers!