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Find the value of: 2x3 – 11x2 + 44x + 27, if x = 253-4i - Mathematics and Statistics

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Question

Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`

Sum
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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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Chapter 3: Complex Numbers - EXERCISE 3.1 [Page 38]

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