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# help

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Express \(\dfrac{30+19i}{4+9i}\) in the form \(a + bi\), where  \(a\)and \(b\) are real numbers.

Sep 29, 2018

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Simplify the following:
(19 i + 30)/(9 i + 4)

Multiply numerator and denominator of (19 i + 30)/(9 i + 4) by the conjugate of the denominator.

Multiply numerator and denominator of (19 i + 30)/(9 i + 4) by 4 - 9 i:
((19 i + 30) (-9 i + 4))/((9 i + 4) (-9 i + 4))

Multiply 4 + 9 i and 4 - 9 i together using FOIL.
(4 + 9 i) (4 - 9 i) = 4×4 + 4 (-9 i) + 9 i×4 + 9 i (-9 i) = 16 - 36 i + 36 i + 81 = 97:
((19 i + 30) (-9 i + 4))/97

Multiply 30 + 19 i and 4 - 9 i together using FOIL.
(30 + 19 i) (4 - 9 i) = 30×4 + 30 (-9 i) + 19 i×4 + 19 i (-9 i) = 120 - 270 i + 76 i + 171 = 291 - 194 i:
(-194 i + 291)/97

Factor common terms from -194 i + 291.
Factor 97 out of 291 - 194 i giving 97 (3 - 2 i):
(97 (-2 i + 3))/97

Cancel common terms in the numerator and denominator of (97 (-2 i + 3))/97.
(97 (-2 i + 3))/97 = 97/97×(-2 i + 3) = -2 i + 3:
-2 i + 3 = 3 - 2i

Sep 29, 2018