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February 27th, 2017, 04:24 AM   #1
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putting in form z=x+iy

Please could some one help with how to put this equation into the form x+iy=z as I am stuck
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February 27th, 2017, 05:04 AM   #2
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$z^2 + 2z - i \sqrt{3} = 0$

complete the square

$(z+1)^2 - 1 - i\sqrt{3} = 0$

$(z+1)^2 = 1+i\sqrt{3}$

$z+1 = \pm \sqrt{1+i\sqrt{3}}$

$z = -1 \pm \sqrt{1+i\sqrt{3}}$

Now we need to deal with the radical.

$1+i\sqrt{3} = 2e^{i\pi/3}$

$\sqrt{1+i\sqrt{3}} = \sqrt{2}e^{i\pi/6} = \sqrt{\dfrac 3 2 }+\dfrac{i}{\sqrt{2}}$

so

$z = -1 + \sqrt{\dfrac 3 2 }+\dfrac{i}{\sqrt{2}}$
$z = -1 - \sqrt{\dfrac 3 2 }-\dfrac{i}{\sqrt{2}}$
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