z^2+(2+i)z+1=0

Solution for z^2+(2+i)z+1=0 equation:

Simplifying
z2 + (2 + i) * z + 1 = 0

Reorder the terms for easier multiplication:
z2 + z(2 + i) + 1 = 0
z2 + (2 * z + i * z) + 1 = 0

Reorder the terms:
z2 + (iz + 2z) + 1 = 0
z2 + (iz + 2z) + 1 = 0

Reorder the terms:
1 + iz + 2z + z2 = 0

Solving
1 + iz + 2z + z2 = 0

Solving for variable 'i'.

Move all terms containing i to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + iz + 2z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + iz + 2z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + iz + 2z + z2 = 0 + -1
iz + 2z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
iz + 2z + z2 = -1

Add '-2z' to each side of the equation.
iz + 2z + -2z + z2 = -1 + -2z

Combine like terms: 2z + -2z = 0
iz + 0 + z2 = -1 + -2z
iz + z2 = -1 + -2z

Add '-1z2' to each side of the equation.
iz + z2 + -1z2 = -1 + -2z + -1z2

Combine like terms: z2 + -1z2 = 0
iz + 0 = -1 + -2z + -1z2
iz = -1 + -2z + -1z2

Divide each side by 'z'.
i = -1z-1 + -2 + -1z

Simplifying
i = -1z-1 + -2 + -1z

Reorder the terms:
i = -2 + -1z-1 + -1z