What is complex z given z+6conjugate z+3i=0?

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You need to consider the complex number `z = x + i*y` , whose conjugate is `bar z = x - i*y` , hence, replacing `x + i*y` for `z` and `x - i*y` for `bar z` in the equation provided by the problem, yields:

`x + i*y + 6(x - i*y) + 3i = 0`

`x + i*y + 6x - 6i*y + 3i = 0`

`7x - 5i*y + 3i = 0`

`7x + i*(-5y + 3) = 0 => 7x + i*(-5y + 3) = 0 + 0*i`

You need to find the parts x and y of the complex number z equating the real parts and imaginary parts both sides, such that:

`7x = 0 => x = 0`

`-5y + 3 = 0 => -5y = -3 => y = 3/5`

Hence, evaluating the complex number z, under the given conditions, yields `z = 0 + (3/5)*i.`

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