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What are modulus of complex roots in z^2+2z+4=0?

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greenbel | (Level 2) Honors

Posted June 25, 2013 at 1:10 PM via web

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What are modulus of complex roots in z^2+2z+4=0?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 25, 2013 at 1:34 PM (Answer #1)

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You need to evaluate the solutions to quadratic equation `z^2 + 2z + 4 = 0` , using quadratic formula, such that:

`z_(1,2) = (-2+-sqrt(4 - 16))/2`

`z_(1,2) = (-2+-sqrt(-12))/2`

Since in complex number theory, `sqrt(-1) = i` , yields:

`z_(1,2) = (-2+-i*sqrt12)/2`

`z_(1,2) = (-2+-2i*sqrt3)/2`

Factoring out 2 yields:

`z_(1,2) = 2(-1+-i*sqrt3)/2`

Reducing duplicate factors yields:

`z_(1,2) = -1+-i*sqrt3`

You should notice that the absolute values of solutions `z_(1,2)` are equal, such that:

`|z_1| = |z_2| => |z| = sqrt((-1)^2 + (sqrt3)^2)`

`|z| = sqrt(1 + 3) => |z| = sqrt4 => |z| = 2`

Hence, evaluating the absolute values of the complex conjugate solutions to the quadratic equation `z^2 + 2z + 4 = 0` yields `|z_1| = |z_2| = 2` .

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