Question

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

Answer 1

( 30 + 19i )/( 4 + 9i)

multiply ( 4 -9i) with numerator and denominator

= (30 + 19i)(4 - 9i)/(4 + 9i)(4 - 9i)

={ 30( 4 -9i) +19i(4 - 9i)}/{4² -(9i)²}


={ 120 -270i + 76i -171(i²)}/{16 - 81(i²)}

we know ,
i² = -1 use this here,

={ 120 -194i +171 }/{ 16 + 81}

= ( 291 - 194i)/97

= (291/97) - (194/97)i

hence, (291/97) + (-194/97)i is in the form of a + bi

Answer 2

Answer:

multiply top/bottm by 4 - 9i

[ 30 + 19i ] [ 4 - 9i ] / ( 4 + 9i) (4 - 9i) ]

[ 120 + 76i - 270i - 171i^2 ] / [ 16 - 81i^2 ]

[ 120 + 171 - 194i ] / [ 16 + 81 ]

[ 291 - 194i ] / [ 97 ]

(291/ 97) - (194/97) i

3 - 2i