Question 1170874
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Let m and n be complex numbers
Furthermore, let n be the complex conjugate of m
This means
m = a+bi
n = a-bi
for real numbers a,b


Let's find an equivalent expression for (4+2i)m
(4+2i)m = (4+2i)(a+bi)
(4+2i)m = 4(a+bi)+2i(a+bi)
(4+2i)m = 4a+4bi+2ai+2bi^2
(4+2i)m = 4a+4bi+2ai+2b(-1)
(4+2i)m = 4a+4bi+2ai-2b
(4+2i) = (4a-2b)+(4bi+2ai)
(4+2i)m = (4a-2b)+(4b+2a)i
I'm using parenthesis to help separate out the real and imaginary parts.


Similarly, let's find (1-i)n
(1-i)n = (1-i)(a-bi)
(1-i)n = 1(a-bi)-i(a-bi)
(1-i)n = a-bi-ai+bi^2
(1-i)n = a-bi-ai+b(-1)
(1-i)n = a-bi-ai-b
(1-i)n = (a-b)+(-ai-bi)
(1-i)n = (a-b)-(ai+bi)
(1-i)n = (a-b)-(a+b)i


This means,
(4+2i)m-(1-i)n = [(4a-2b)+(4b+2a)i] - [(a-b)-(a+b)i]
(4+2i)m-(1-i)n = (4a-2b)+(4b+2a)i - (a-b)+(a+b)i
(4+2i)m-(1-i)n = [(4a-2b) - (a-b)] + [(4b+2a)i+(a+b)i]
(4+2i)m-(1-i)n = [(4a-2b) - (a-b)] + [(4b+2a)+(a+b)]i
(4+2i)m-(1-i)n = (4a-2b - a+b) + (4b+2a+a+b)i
(4+2i)m-(1-i)n = (3a-b) + (3a+5b)i



Equate that result with -9+9i
(3a-b) + (3a+5b)i = -9+9i
The corresponding parts must match up and be equal
<ul><li>3a-b = -9</li><li>(3a+5b)i = 9i which leads to 3a+5b = 9</li></ul>
The system of equations is
3a-b = -9
3a+5b = 9


Let's subtract the terms straight down
3a-3a = 0a
-b-5b = -6b
-9-9 = -18


We end up with -6b = -18 which solves to b = 3 after dividing both sides by -6.


We can then find 'a'
3a-b = -9
3a-3 = -9
3a = -9+3
3a = -6
a = -6/3
a = -2


In summary we found that
a = -2
b = 3


which directly leads to
m = a+bi = -2+3i
n = a-bi = -2-3i


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Checking the answers:


(4+2i)m = (4+2i)(-2+3i)
(4+2i)m = 4(-2+3i)+2i(-2+3i)
(4+2i)m = -8+12i-4i+6i^2
(4+2i)m = -8+12i-4i+6(-1)
(4+2i)m = -8+12i-4i-6
(4+2i)m = (-8-6)+(12-4)i
(4+2i)m = -14+8i


(1-i)n = (1-i)(-2-3i)
(1-i)n = 1(-2-3i)-i(-2-3i)
(1-i)n = -2-3i+2i+3i^2
(1-i)n = -2-3i+2i+3(-1)
(1-i)n = -2-3i+2i-3
(1-i)n = (-2-3)+(-3+2)i
(1-i)n = -5-i


(4+2i)m-(1-i)n = (-14+8i) - (-5-i)
(4+2i)m-(1-i)n = -14+8i + 5+i
(4+2i)m-(1-i)n = (-14+5)+(8i+i)
(4+2i)m-(1-i)n = (-14+5)+(8+1)i
(4+2i)m-(1-i)n = -9+9i


The answers have been confirmed.

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Answers:
m = -2+3i
n = -2-3i
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