Divide 5 + -7i and 7 + 3i

a = bi = <-- Enter a and bi piece
c = di = <-- Enter c and di piece (not needed for square root or absolute value or conjugate)

Perform the complex number division below:
5 - 7i
7 + 3i

Complex number division involves multiplying numerator and denominator by the conjugate of the denominator.
If the denominator is c + di, the conjugate is c - di. Multiplying top and bottom by the conjugate (7 - 3i), we get:
(5 - 7i)(7 - 3i)
(7 + 3i)(7 - 3i)

Expand the denominator (7 + 3i)(7 - 3i)
The formula for this using the FOIL method is: (a * c) + (b * c) + (a * d) + (b * d) where:
a = 7, b = 3, c = 7, and d = -3

Now plug these values into our formula and evaluate:
(7 + 3i)(7 - 3i) = (7 * 7) + (3i * 7) + (7 * -3i) + (3i * -3i)
(7 + 3i)(7 - 3i) = 49 + 21i - 21i - 9i2

Group the like terms that contain i:
(7 + 3i)(7 - 3i) = 49 + (21 - 21)i - 9i2
(7 + 3i)(7 - 3i) = 49 - 9i2

Simplify our last term:
i2 = √-1 * √-1 = -1, so our last term becomes:
(7 + 3i)(7 - 3i) = 49 - 9* (-1)
(7 + 3i)(7 - 3i) = 49 + 9

Now group the 2 constants and finalize our answer
(7 + 3i)(7 - 3i) = (49 + 9)
(7 + 3i)(7 - 3i) = 58

Expand the numerator (5 - 7i)(7 - 3i)
The formula for this using the FOIL method is: (a * c) + (b * c) + (a * d) + (b * d) where:
a = 5, b = -7, c = 7, and d = -3

Now plug these values into our formula and evaluate:
(5 - 7i)(7 - 3i) = (5 * 7) + (-7i * 7) + (5 * -3i) + (-7i * -3i)
(5 - 7i)(7 - 3i) = 35 - 49i - 15i + 21i2

Group the like terms that contain i:
(5 - 7i)(7 - 3i) = 35 + (-49 - 15)i + 21i2
(5 - 7i)(7 - 3i) = 35 - 64i + 21i2

Simplify our last term:
i2 = √-1 * √-1 = -1, so our last term becomes:
(5 - 7i)(7 - 3i) = 35 - 64i + 21* (-1)
(5 - 7i)(7 - 3i) = 35 - 64i - 21

Now group the 2 constants and finalize our answer
(5 - 7i)(7 - 3i) = (35 - 21) - 64i
(5 - 7i)(7 - 3i) = 14 - 64i

After expanding and simplifying numerator and denominator, we are left with:
5 - 7i
7 + 3i
=
14 - 64i
58

Our fraction is not fully reduced. The Greatest Common Factor (GCF) of 14, -64, and 58 is 2. Reducing our fraction by the GCF, we get our answer:
5 - 7i
7 + 3i
=
7 - 32i
29


Answer
5 - 7i
7 + 3i
=
7 - 32i
29