(a+bi)(g-hi)=(hi+g)(a+bi)

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Solution for (a+bi)(g-hi)=(hi+g)(a+bi) equation: 


Simplifying
(a + bi)(g + -1hi) = (hi + g)(a + bi)

Multiply (a + bi) * (g + -1hi)
(a(g + -1hi) + bi(g + -1hi)) = (hi + g)(a + bi)
((g * a + -1hi * a) + bi(g + -1hi)) = (hi + g)(a + bi)
((ag + -1ahi) + bi(g + -1hi)) = (hi + g)(a + bi)
(ag + -1ahi + (g * bi + -1hi * bi)) = (hi + g)(a + bi)
(ag + -1ahi + (bgi + -1bhi2)) = (hi + g)(a + bi)
(ag + -1ahi + bgi + -1bhi2) = (hi + g)(a + bi)

Reorder the terms:
ag + -1ahi + bgi + -1bhi2 = (g + hi)(a + bi)

Multiply (g + hi) * (a + bi)
ag + -1ahi + bgi + -1bhi2 = (g(a + bi) + hi(a + bi))
ag + -1ahi + bgi + -1bhi2 = ((a * g + bi * g) + hi(a + bi))
ag + -1ahi + bgi + -1bhi2 = ((ag + bgi) + hi(a + bi))
ag + -1ahi + bgi + -1bhi2 = (ag + bgi + (a * hi + bi * hi))
ag + -1ahi + bgi + -1bhi2 = (ag + bgi + (ahi + bhi2))

Reorder the terms:
ag + -1ahi + bgi + -1bhi2 = (ag + ahi + bgi + bhi2)
ag + -1ahi + bgi + -1bhi2 = (ag + ahi + bgi + bhi2)

Add '-1ag' to each side of the equation.
ag + -1ahi + bgi + -1ag + -1bhi2 = ag + ahi + bgi + -1ag + bhi2

Reorder the terms:
ag + -1ag + -1ahi + bgi + -1bhi2 = ag + ahi + bgi + -1ag + bhi2

Combine like terms: ag + -1ag = 0
0 + -1ahi + bgi + -1bhi2 = ag + ahi + bgi + -1ag + bhi2
-1ahi + bgi + -1bhi2 = ag + ahi + bgi + -1ag + bhi2

Reorder the terms:
-1ahi + bgi + -1bhi2 = ag + -1ag + ahi + bgi + bhi2

Combine like terms: ag + -1ag = 0
-1ahi + bgi + -1bhi2 = 0 + ahi + bgi + bhi2
-1ahi + bgi + -1bhi2 = ahi + bgi + bhi2

Add '-1bgi' to each side of the equation.
-1ahi + bgi + -1bgi + -1bhi2 = ahi + bgi + -1bgi + bhi2

Combine like terms: bgi + -1bgi = 0
-1ahi + 0 + -1bhi2 = ahi + bgi + -1bgi + bhi2
-1ahi + -1bhi2 = ahi + bgi + -1bgi + bhi2

Combine like terms: bgi + -1bgi = 0
-1ahi + -1bhi2 = ahi + 0 + bhi2
-1ahi + -1bhi2 = ahi + bhi2

Solving
-1ahi + -1bhi2 = ahi + bhi2

Solving for variable 'a'.

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

Add '-1ahi' to each side of the equation.
-1ahi + -1ahi + -1bhi2 = ahi + -1ahi + bhi2

Combine like terms: -1ahi + -1ahi = -2ahi
-2ahi + -1bhi2 = ahi + -1ahi + bhi2

Combine like terms: ahi + -1ahi = 0
-2ahi + -1bhi2 = 0 + bhi2
-2ahi + -1bhi2 = bhi2

Add 'bhi2' to each side of the equation.
-2ahi + -1bhi2 + bhi2 = bhi2 + bhi2

Combine like terms: -1bhi2 + bhi2 = 0
-2ahi + 0 = bhi2 + bhi2
-2ahi = bhi2 + bhi2

Combine like terms: bhi2 + bhi2 = 2bhi2
-2ahi = 2bhi2

Divide each side by '-2hi'.
a = -1bi

Simplifying
a = -1bi

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