(3-5i)(2+9i)+(7+2i)(7-2i)=

Solution for (3-5i)(2+9i)+(7+2i)(7-2i)= equation:

Simplifying
(3 + -5i)(2 + 9i) + (7 + 2i)(7 + -2i) = 0

Multiply (3 + -5i) * (2 + 9i)
(3(2 + 9i) + -5i * (2 + 9i)) + (7 + 2i)(7 + -2i) = 0
((2 * 3 + 9i * 3) + -5i * (2 + 9i)) + (7 + 2i)(7 + -2i) = 0
((6 + 27i) + -5i * (2 + 9i)) + (7 + 2i)(7 + -2i) = 0
(6 + 27i + (2 * -5i + 9i * -5i)) + (7 + 2i)(7 + -2i) = 0
(6 + 27i + (-10i + -45i2)) + (7 + 2i)(7 + -2i) = 0

Combine like terms: 27i + -10i = 17i
(6 + 17i + -45i2) + (7 + 2i)(7 + -2i) = 0

Multiply (7 + 2i) * (7 + -2i)
6 + 17i + -45i2 + (7(7 + -2i) + 2i * (7 + -2i)) = 0
6 + 17i + -45i2 + ((7 * 7 + -2i * 7) + 2i * (7 + -2i)) = 0
6 + 17i + -45i2 + ((49 + -14i) + 2i * (7 + -2i)) = 0
6 + 17i + -45i2 + (49 + -14i + (7 * 2i + -2i * 2i)) = 0
6 + 17i + -45i2 + (49 + -14i + (14i + -4i2)) = 0

Combine like terms: -14i + 14i = 0
6 + 17i + -45i2 + (49 + 0 + -4i2) = 0
6 + 17i + -45i2 + (49 + -4i2) = 0

Reorder the terms:
6 + 49 + 17i + -45i2 + -4i2 = 0

Combine like terms: 6 + 49 = 55
55 + 17i + -45i2 + -4i2 = 0

Combine like terms: -45i2 + -4i2 = -49i2
55 + 17i + -49i2 = 0

Solving
55 + 17i + -49i2 = 0

Solving for variable 'i'.

Begin completing the square.  Divide all terms by
-49 the coefficient of the squared term: 

Divide each side by '-49'.
-1.12244898 + -0.3469387755i + i2 = 0

Move the constant term to the right:

Add '1.12244898' to each side of the equation.
-1.12244898 + -0.3469387755i + 1.12244898 + i2 = 0 + 1.12244898

Reorder the terms:
-1.12244898 + 1.12244898 + -0.3469387755i + i2 = 0 + 1.12244898

Combine like terms: -1.12244898 + 1.12244898 = 0.00000000
0.00000000 + -0.3469387755i + i2 = 0 + 1.12244898
-0.3469387755i + i2 = 0 + 1.12244898

Combine like terms: 0 + 1.12244898 = 1.12244898
-0.3469387755i + i2 = 1.12244898

The i term is -0.3469387755i.  Take half its coefficient (-0.1734693878).
Square it (0.03009162850) and add it to both sides.

Add '0.03009162850' to each side of the equation.
-0.3469387755i + 0.03009162850 + i2 = 1.12244898 + 0.03009162850

Reorder the terms:
0.03009162850 + -0.3469387755i + i2 = 1.12244898 + 0.03009162850

Combine like terms: 1.12244898 + 0.03009162850 = 1.1525406085
0.03009162850 + -0.3469387755i + i2 = 1.1525406085

Factor a perfect square on the left side:
(i + -0.1734693878)(i + -0.1734693878) = 1.1525406085

Calculate the square root of the right side: 1.073564441

Break this problem into two subproblems by setting 
(i + -0.1734693878) equal to 1.073564441 and -1.073564441.

Subproblem 1
i + -0.1734693878 = 1.073564441

Simplifying
i + -0.1734693878 = 1.073564441

Reorder the terms:
-0.1734693878 + i = 1.073564441

Solving
-0.1734693878 + i = 1.073564441

Solving for variable 'i'.

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

Add '0.1734693878' to each side of the equation.
-0.1734693878 + 0.1734693878 + i = 1.073564441 + 0.1734693878

Combine like terms: -0.1734693878 + 0.1734693878 = 0.0000000000
0.0000000000 + i = 1.073564441 + 0.1734693878
i = 1.073564441 + 0.1734693878

Combine like terms: 1.073564441 + 0.1734693878 = 1.2470338288
i = 1.2470338288

Simplifying
i = 1.2470338288

Subproblem 2
i + -0.1734693878 = -1.073564441

Simplifying
i + -0.1734693878 = -1.073564441

Reorder the terms:
-0.1734693878 + i = -1.073564441

Solving
-0.1734693878 + i = -1.073564441

Solving for variable 'i'.

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

Add '0.1734693878' to each side of the equation.
-0.1734693878 + 0.1734693878 + i = -1.073564441 + 0.1734693878

Combine like terms: -0.1734693878 + 0.1734693878 = 0.0000000000
0.0000000000 + i = -1.073564441 + 0.1734693878
i = -1.073564441 + 0.1734693878

Combine like terms: -1.073564441 + 0.1734693878 = -0.9000950532
i = -0.9000950532

Simplifying
i = -0.9000950532

Solution
The solution to the problem is based on the solutions
from the subproblems.
i = {1.2470338288, -0.9000950532}