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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:
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"i")`
Solution
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"i")`
= `((3 + 2"i")(2 + 5"i") + (2 - 5"i")(3 -2"i"))/((2 - 5"i")(2 + 5"i"))`
= `(6 + 15"i" + 4"i" + 10"i"^2 + 6 - 4"i" - 15"i" + 10"i"^2)/(4 - 25"i"^2)`
= `(12 + 20"i"^2)/(4 - 25"i"^2)`
= `(12 + 20(-1))/(4 -25(-1))` ...[∵ i2 = – 1]
= `(-8)/29`
∴ `(3 + 2"i")/(2 - 5"i") + (3 - 2"i")/(2 + 5"i") = (-8)/29 + 0"i"`
∴ a = `(-8)/29` and b = 0
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