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1) For how many real values of c do we have $$\left|\frac12-ci\right| = \frac34$$?

2) Determine the complex number z satisfying the equation $$2z-3\bar{z}=-2-30i$$.

Aug 9, 2019

#1
+6203
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$$\left|\dfrac 1 2 - c i \right| = \left(\dfrac 1 2 \right)^2 + c^2 = \dfrac 3 4\\ \dfrac 1 4 + c^2 = \dfrac 3 4\\ c^2 = \dfrac 1 2\\ c = \pm \dfrac{1}{\sqrt{2}}$$

$$2z - 3\bar{z} = -2 - 30i\\ \text{let z=x+iy}\\ 2x + i2y - 3x + i3y = -2 - 30i\\ -x+i5y=-2-30i\\ x = 2,~y = -6i\\ z = 2-6i$$

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Aug 9, 2019
edited by Rom  Aug 9, 2019
#2
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Bad day at the office Rom ?

Guest Aug 9, 2019
#4
+6203
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any day at an office would be a bad day.

Rom  Aug 9, 2019
#5
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It's just an expression, don't take it literally.

You need to look at Q1 as well.

Guest Aug 9, 2019
#3
+114141
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2) Let  z  =  a + bi

So we have

2 (a + bi)  - 3 (a - bi)   =  -2  - 30i      simplify

2a + 2bi   - 3a  + 3bi  =   -2  - 30i

-1a + 5bi =   -2 - 30i

Equate terms

-1a   = -2             5bi  = -30i

a  =  2           b  = -6

So

z =  2  -  6i

Aug 9, 2019