How do I simplify this?

#(4+2i)/(-4-6i)#

2 Answers
Mr. Hsia
May 16, 2018

#(-1 + 4i) / 13#

Explanation:

Multiply the Numerator and Denominator by the Conjugate of the Denominator.
The Conjugate is exactly like the Denominator except the sign on the imaginary number is different.
#(4+2i)/(-4-6i) * (-4+6i)/(-4 + 6i)#

Multiply by Distribution on Numerator and Denominator.
# = (-16 + 24i - 8i - 12i^2)/(16 - 24i + 24i - 36i^2)#

Note: #i = sqrt(-1)# so #i^2 = -1#
Combine Like Terms and Replace #i^2# with #-1#
# = (-16 + 16i - 12(-1)) / (16 - 36(-1))#
# = (-16 + 16i + 12) / (16 + 36) #
# = (-4 + 16i) / 52 #

Simplify by Dividing all Numbers by 4:
# = (-1 + 4i) / 13#

A08
May 16, 2018

Given

#(4+2i)/(-4-6i)#

From inspection we see that both in the numerator and denominator factor #2# can be taken out as common. This cancels out. Therefore, we get

#-(2(2+i))/(2(2+3i))#
#=>-((2+i))/((2+3i))#

Multiplying and dividing with complex conjugate of denominator we get

#-((2+i))/((2+3i))xx(2-3i)/(2-3i)#
#=>-((2+i)(2-3i))/((2)^2-(3i)^2)#
#=>-(4-6i+2i-3i^2)/((2)^2-(3i)^2)#
#=>-(7-4i)/(13)#
#=>(4i-7)/(13)#

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