(7+8i)(3-i)-(1-i)(1+i)=

Solution for (7+8i)(3-i)-(1-i)(1+i)= equation:

Simplifying
(7 + 8i)(3 + -1i) + -1(1 + -1i)(1 + i) = 0

Multiply (7 + 8i) * (3 + -1i)
(7(3 + -1i) + 8i * (3 + -1i)) + -1(1 + -1i)(1 + i) = 0
((3 * 7 + -1i * 7) + 8i * (3 + -1i)) + -1(1 + -1i)(1 + i) = 0
((21 + -7i) + 8i * (3 + -1i)) + -1(1 + -1i)(1 + i) = 0
(21 + -7i + (3 * 8i + -1i * 8i)) + -1(1 + -1i)(1 + i) = 0
(21 + -7i + (24i + -8i2)) + -1(1 + -1i)(1 + i) = 0

Combine like terms: -7i + 24i = 17i
(21 + 17i + -8i2) + -1(1 + -1i)(1 + i) = 0

Multiply (1 + -1i) * (1 + i)
21 + 17i + -8i2 + -1(1(1 + i) + -1i * (1 + i)) = 0
21 + 17i + -8i2 + -1((1 * 1 + i * 1) + -1i * (1 + i)) = 0
21 + 17i + -8i2 + -1((1 + 1i) + -1i * (1 + i)) = 0
21 + 17i + -8i2 + -1(1 + 1i + (1 * -1i + i * -1i)) = 0
21 + 17i + -8i2 + -1(1 + 1i + (-1i + -1i2)) = 0

Combine like terms: 1i + -1i = 0
21 + 17i + -8i2 + -1(1 + 0 + -1i2) = 0
21 + 17i + -8i2 + -1(1 + -1i2) = 0
21 + 17i + -8i2 + (1 * -1 + -1i2 * -1) = 0
21 + 17i + -8i2 + (-1 + 1i2) = 0

Reorder the terms:
21 + -1 + 17i + -8i2 + 1i2 = 0

Combine like terms: 21 + -1 = 20
20 + 17i + -8i2 + 1i2 = 0

Combine like terms: -8i2 + 1i2 = -7i2
20 + 17i + -7i2 = 0

Solving
20 + 17i + -7i2 = 0

Solving for variable 'i'.

Begin completing the square.  Divide all terms by
-7 the coefficient of the squared term: 

Divide each side by '-7'.
-2.857142857 + -2.428571429i + i2 = 0

Move the constant term to the right:

Add '2.857142857' to each side of the equation.
-2.857142857 + -2.428571429i + 2.857142857 + i2 = 0 + 2.857142857

Reorder the terms:
-2.857142857 + 2.857142857 + -2.428571429i + i2 = 0 + 2.857142857

Combine like terms: -2.857142857 + 2.857142857 = 0.000000000
0.000000000 + -2.428571429i + i2 = 0 + 2.857142857
-2.428571429i + i2 = 0 + 2.857142857

Combine like terms: 0 + 2.857142857 = 2.857142857
-2.428571429i + i2 = 2.857142857

The i term is -2.428571429i.  Take half its coefficient (-1.214285715).
Square it (1.474489798) and add it to both sides.

Add '1.474489798' to each side of the equation.
-2.428571429i + 1.474489798 + i2 = 2.857142857 + 1.474489798

Reorder the terms:
1.474489798 + -2.428571429i + i2 = 2.857142857 + 1.474489798

Combine like terms: 2.857142857 + 1.474489798 = 4.331632655
1.474489798 + -2.428571429i + i2 = 4.331632655

Factor a perfect square on the left side:
(i + -1.214285715)(i + -1.214285715) = 4.331632655

Calculate the square root of the right side: 2.08125747

Break this problem into two subproblems by setting 
(i + -1.214285715) equal to 2.08125747 and -2.08125747.

Subproblem 1
i + -1.214285715 = 2.08125747

Simplifying
i + -1.214285715 = 2.08125747

Reorder the terms:
-1.214285715 + i = 2.08125747

Solving
-1.214285715 + i = 2.08125747

Solving for variable 'i'.

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

Add '1.214285715' to each side of the equation.
-1.214285715 + 1.214285715 + i = 2.08125747 + 1.214285715

Combine like terms: -1.214285715 + 1.214285715 = 0.000000000
0.000000000 + i = 2.08125747 + 1.214285715
i = 2.08125747 + 1.214285715

Combine like terms: 2.08125747 + 1.214285715 = 3.295543185
i = 3.295543185

Simplifying
i = 3.295543185

Subproblem 2
i + -1.214285715 = -2.08125747

Simplifying
i + -1.214285715 = -2.08125747

Reorder the terms:
-1.214285715 + i = -2.08125747

Solving
-1.214285715 + i = -2.08125747

Solving for variable 'i'.

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

Add '1.214285715' to each side of the equation.
-1.214285715 + 1.214285715 + i = -2.08125747 + 1.214285715

Combine like terms: -1.214285715 + 1.214285715 = 0.000000000
0.000000000 + i = -2.08125747 + 1.214285715
i = -2.08125747 + 1.214285715

Combine like terms: -2.08125747 + 1.214285715 = -0.866971755
i = -0.866971755

Simplifying
i = -0.866971755

Solution
The solution to the problem is based on the solutions
from the subproblems.
i = {3.295543185, -0.866971755}