(a+bi)(2+i)=a-3i

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Solution for (a+bi)(2+i)=a-3i equation: 


Simplifying
(a + bi)(2 + i) = a + -3i

Multiply (a + bi) * (2 + i)
(a(2 + i) + bi(2 + i)) = a + -3i
((2 * a + i * a) + bi(2 + i)) = a + -3i
((2a + ai) + bi(2 + i)) = a + -3i
(2a + ai + (2 * bi + i * bi)) = a + -3i
(2a + ai + (2bi + bi2)) = a + -3i
(2a + ai + 2bi + bi2) = a + -3i

Solving
2a + ai + 2bi + bi2 = a + -3i

Solving for variable 'a'.

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

Add '-1a' to each side of the equation.
2a + ai + 2bi + -1a + bi2 = a + -1a + -3i

Reorder the terms:
2a + -1a + ai + 2bi + bi2 = a + -1a + -3i

Combine like terms: 2a + -1a = 1a
1a + ai + 2bi + bi2 = a + -1a + -3i

Combine like terms: a + -1a = 0
1a + ai + 2bi + bi2 = 0 + -3i
1a + ai + 2bi + bi2 = -3i

Add '-2bi' to each side of the equation.
1a + ai + 2bi + -2bi + bi2 = -2bi + -3i

Combine like terms: 2bi + -2bi = 0
1a + ai + 0 + bi2 = -2bi + -3i
1a + ai + bi2 = -2bi + -3i

Add '-1bi2' to each side of the equation.
1a + ai + bi2 + -1bi2 = -2bi + -1bi2 + -3i

Combine like terms: bi2 + -1bi2 = 0
1a + ai + 0 = -2bi + -1bi2 + -3i
1a + ai = -2bi + -1bi2 + -3i

Reorder the terms:
1a + ai + 2bi + bi2 + 3i = -2bi + 2bi + -1bi2 + bi2 + -3i + 3i

Combine like terms: -2bi + 2bi = 0
1a + ai + 2bi + bi2 + 3i = 0 + -1bi2 + bi2 + -3i + 3i
1a + ai + 2bi + bi2 + 3i = -1bi2 + bi2 + -3i + 3i

Combine like terms: -1bi2 + bi2 = 0
1a + ai + 2bi + bi2 + 3i = 0 + -3i + 3i
1a + ai + 2bi + bi2 + 3i = -3i + 3i

Combine like terms: -3i + 3i = 0
1a + ai + 2bi + bi2 + 3i = 0

The solution to this equation could not be determined.

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