𝑧2 + 6𝑧 + 25 = 0 and Express the solutions in the form 𝑎 + 𝑏𝑖.

Solution:
1. Identify the coefficients in the quadratic equation z^2 + 6z + 25 = 0:
- a = 1, b = 6, c = 25.

2. Use the quadratic formula z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} to find the solutions.

3. Calculate the discriminant b^2 - 4ac:
- b^2 = 6^2 = 36
- 4ac = 4 \times 1 \times 25 = 100
- Discriminant: b^2 - 4ac = 36 - 100 = -64

4. Substitute back into the quadratic formula:
z = \frac{-6 \pm \sqrt{-64}}{2 \cdot 1}

5. Simplify:
- The square root of -64 is 8i (since \sqrt{-64} = 8i.
- Therefore, z = \frac{-6 \pm 8i}{2}.

6. Split the expression for each solution:
- z_1 = \frac{-6 + 8i}{2} = -3 + 4i
- z_2 = \frac{-6 - 8i}{2} = -3 - 4i

7. The solutions in the form a + bi are:
- z_1 = -3 + 4i
- z_2 = -3 - 4i