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Question
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
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Solution 1
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
`= (2 + "i")/(3 + 6"i" - "i" - 2"i"^2)`
= `(2 + "i")/(3 + 5"i" - 2(-1))` ...[∵ i2 = – 1]
= `(2 + "i")/(5 + 5"i")`
= `(2 + "i")/(5(1 + "i"))`
= `((2 + "i")(1 - "i"))/(5(1 + "i")(1 - "i"))`
= `(2 - 2"i" + "i" - "i"^2)/(5(1 - "i"^2)`
= `(2 - "i" - (-1))/(5[1 - (-1)]` ...[∵ i2 = – 1]
= `(3- "i")/10`
∴ `(2 + "i")/((3 - "i")(1 + 2"i")) = 3/10 - 1/10"i"`
∴ a = `3/10` and b = `(-1)/10`
Solution 2
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
`= (2 + "i")/(3 + 6"i" - "i" - 2"i"^2)`
= `(2 + "i")/(3 + 5"i" - 2(-1))` ...[∵ i2 = – 1]
= `(2 + "i")/(3 + 5"i" + 2)`
= `(2 + "i")/(5 + 5"i")`
= `((2 + "i").(5 - 5"i"))/((5 + 5"i").(5 - 5"i"))`
= `(10 - 10"i" + 5"i" - 5"i"^2)/(5^2 - 5"i"^2)`
= `(10 - 10"i" + 5"i" - 5(-1))/(5^2 - 5"i"^2)` ...[∵ i2 = – 1]
= `(10 - 10"i" + 5"i" + 5)/(5^2 - 5"i"^2)`
= `(15 - 5"i")/(25 - 25(-1))`
= `(15 - 5"i")/(25 +25)`
= `(15 - 5"i")/(50)`
= `15/50 - (5"i")/50`
= `3/10 - (1"i")/10`
∴ write in a + ib form
∴ a = `3/10` and b = `(-1)/10`
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