abs(a+bi+3i)=3abs(a+bi)

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Solution for abs(a+bi+3i)=3abs(a+bi) equation: 


Simplifying
abs(a + bi + 3i) = 3abs(a + bi)
(a * abs + bi * abs + 3i * abs) = 3abs(a + bi)

Reorder the terms:
(3abis + ab2is + a2bs) = 3abs(a + bi)
(3abis + ab2is + a2bs) = 3abs(a + bi)
3abis + ab2is + a2bs = (a * 3abs + bi * 3abs)

Reorder the terms:
3abis + ab2is + a2bs = (3ab2is + 3a2bs)
3abis + ab2is + a2bs = (3ab2is + 3a2bs)

Solving
3abis + ab2is + a2bs = 3ab2is + 3a2bs

Solving for variable 'a'.

Reorder the terms:
3abis + ab2is + -3ab2is + a2bs + -3a2bs = 3ab2is + 3a2bs + -3ab2is + -3a2bs

Combine like terms: ab2is + -3ab2is = -2ab2is
3abis + -2ab2is + a2bs + -3a2bs = 3ab2is + 3a2bs + -3ab2is + -3a2bs

Combine like terms: a2bs + -3a2bs = -2a2bs
3abis + -2ab2is + -2a2bs = 3ab2is + 3a2bs + -3ab2is + -3a2bs

Reorder the terms:
3abis + -2ab2is + -2a2bs = 3ab2is + -3ab2is + 3a2bs + -3a2bs

Combine like terms: 3ab2is + -3ab2is = 0
3abis + -2ab2is + -2a2bs = 0 + 3a2bs + -3a2bs
3abis + -2ab2is + -2a2bs = 3a2bs + -3a2bs

Combine like terms: 3a2bs + -3a2bs = 0
3abis + -2ab2is + -2a2bs = 0

Factor out the Greatest Common Factor (GCF), 'abs'.
abs(3i + -2bi + -2a) = 0


Subproblem 1
Set the factor 'abs' equal to zero and attempt to solve:

Simplifying
abs = 0

Solving
abs = 0

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

Simplifying
abs = 0

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.

Subproblem 2
Set the factor '(3i + -2bi + -2a)' equal to zero and attempt to solve:

Simplifying
3i + -2bi + -2a = 0

Reorder the terms:
-2a + -2bi + 3i = 0

Solving
-2a + -2bi + 3i = 0

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

Add '2bi' to each side of the equation.
-2a + -2bi + 2bi + 3i = 0 + 2bi

Combine like terms: -2bi + 2bi = 0
-2a + 0 + 3i = 0 + 2bi
-2a + 3i = 0 + 2bi
Remove the zero:
-2a + 3i = 2bi

Add '-3i' to each side of the equation.
-2a + 3i + -3i = 2bi + -3i

Combine like terms: 3i + -3i = 0
-2a + 0 = 2bi + -3i
-2a = 2bi + -3i

Divide each side by '-2'.
a = -1bi + 1.5i

Simplifying
a = -1bi + 1.5i
Solution
a = {-1bi + 1.5i}

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