The problem in the image asks to solve the following equation:

$z^2 + 3z + 4 + 10i = 0$

The task is to find the real and imaginary parts of the roots of this equation. Let’s go through the solution:

Step 1: Rewrite the Equation

Rewrite the equation in a more familiar form for a complex variable $z = x + yi$, where $x$ and $y$ are the real and imaginary parts of $z$, respectively.

$z^2 + 3z + (4 + 10i) = 0$

Step 2: Let $z = x + yi$

If we assume $z = x + yi$, then we substitute $z$ in the equation and separate real and imaginary parts. But a simpler approach is to directly use the quadratic formula for complex coefficients.

Step 3: Use the Quadratic Formula

The equation is of the form $az^2 + bz + c = 0$, where:

• $a = 1$,
• $b = 3$,
• $c = 4 + 10i$.

The roots $z$ are given by the quadratic formula:

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

Step 4: Calculate $b^2 - 4ac$

1. $b^2 = 3^2 = 9$.
2. $4ac = 4 \cdot 1 \cdot (4 + 10i) = 16 + 40i$.

So,

$b^2 - 4ac = 9 - (16 + 40i) = -7 - 40i$

Step 5: Find $\sqrt{-7 - 40i}$

To find the square root of $-7 - 40i$, we use polar form. Let $-7 - 40i = r \text{cis} \theta$, where:

• $r = \sqrt{(-7)^2 + (-40)^2} = \sqrt{49 + 1600} = \sqrt{1649}$,
• $\theta = \tan^{-1} \left( \frac{-40}{-7} \right) \approx -99.62^\circ$.

Now, taking the square root gives:

$\sqrt{-7 - 40i} = \sqrt{\sqrt{1649}} \text{cis} \left( \frac{-99.62^\circ}{2} \right)$

You can convert this result back to rectangular form to complete the calculation of the roots.

Would you like to proceed with detailed calculation, or have further questions?

Here are related questions:

1. How do you convert a complex number from rectangular to polar form?
2. What are the steps for calculating the square root of a complex number?
3. How does the quadratic formula apply to equations with complex coefficients?
4. Why does finding roots of complex equations sometimes involve polar coordinates?
5. What are other methods for solving quadratic equations with complex numbers?

Tip: When solving complex equations, isolating real and imaginary components can sometimes simplify your work.