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Question
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
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Solution
x = `25/(3 - 4"i")`
∴ x = `(25(3 + 4"i"))/((3 - 4"i")(3 + 4"i")`
= `(25(3 + 4"i"))/(9 - 16"i"^2)`
= `(25(3 + 4"i"))/(9 - 16(-1))` ...[∵ i2 = – 1]
= `(25(3 + 4"i"))/25`
∴ x = 3 + 4
∴ x – 3 = 4i
∴ (x – 3)2 = 16i2
∴ x2 – 6x + 9 = 16(– 1) ...[∵ i2 = – 1]
∴ x2 – 6x + 25 = 0
2x + 1
`x^2 – 6x + 25")"overline(2x^3 - 11x^2 + 44x + 27)"`
2x3 – 12x2 + 50x
– + –
x2 – 6x + 27
x2 – 6x + 25
– + –
2
∴ 2x3 – 11x2 + 44x + 27
= (x2 – 6x + 25) (2x + 1) + 2
= 0.(2x + 1) + 2 ...[From (i)]
= 0 + 2
= 2
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