The exercise asks to solve the quadratic equation in the complex number set $\mathbb{C}$:

$z^2 - (3 + 4i)z - 1 + 5i = 0$

and to provide the solutions in their simplified algebraic form, then represent them in the Cartesian plane.

Steps to Solve:

1. Standard Form:
   First, express the equation in standard quadratic form: $z^2 - (3 + 4i)z + (-1 + 5i) = 0$ This is a quadratic equation of the form $az^2 + bz + c = 0$, where: $a = 1, \quad b = -(3 + 4i), \quad c = -1 + 5i$

2. Quadratic Formula:
   The solutions to a quadratic equation are given by the quadratic formula: $z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ In this case: $b = -(3 + 4i), \quad a = 1, \quad c = -1 + 5i$ So, substitute these values into the quadratic formula: $z = \frac{(3 + 4i) \pm \sqrt{(3 + 4i)^2 - 4(1)(-1 + 5i)}}{2}$

3. Simplify the Discriminant $\Delta = b^2 - 4ac$:

   • First, calculate $(3 + 4i)^2$: $(3 + 4i)^2 = 9 + 24i - 16 = -7 + 24i$

   • Next, calculate $4ac = 4 \times (1) \times (-1 + 5i) = -4 + 20i$.

   • Now, calculate the discriminant: $\Delta = (-7 + 24i) - (-4 + 20i) = -7 + 24i + 4 - 20i = -3 + 4i$

4. Find the Square Root of the Discriminant $\sqrt{-3 + 4i}$:
   To find the square root of a complex number, let $w = \sqrt{-3 + 4i}$, where $w = x + yi$. Solving this requires using polar or algebraic methods, which leads to: $\sqrt{-3 + 4i} = 1 + 2i$

5. Calculate the Two Solutions:
   Substituting this back into the quadratic formula: $z_1 = \frac{3 + 4i + (1 + 2i)}{2} = \frac{4 + 6i}{2} = 2 + 3i$ $z_2 = \frac{3 + 4i - (1 + 2i)}{2} = \frac{2 + 2i}{2} = 1 + i$

Final Solutions:

The two solutions to the equation are: $z_1 = 2 + 3i, \quad z_2 = 1 + i$

Graphical Representation:

You would now plot these points on the Cartesian plane where:

• $z_1 = 2 + 3i$ corresponds to the point $(2, 3)$.
• $z_2 = 1 + i$ corresponds to the point $(1, 1)$.

Would you like more details or any clarifications on these steps?

Follow-up Questions:

1. How do you find the square root of a complex number algebraically?
2. What is the geometric interpretation of the solutions in the complex plane?
3. How can the quadratic formula be used for any complex quadratic equation?
4. Can you explain the significance of the discriminant in complex quadratic equations?
5. How would you graph other types of complex numbers in the Cartesian plane?

Tip:

When solving complex quadratic equations, always make sure to simplify the discriminant carefully, as it may involve finding the square root of a complex number!