Algebra of complex numbers

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Question

${If x=a+ib is a complex number such that x}^{2} = 3 + 4 i a n d x^{3} = 2 + 11 i w h e r e i = \sqrt{- 1}, t h e n (a + b) =$

Easy

Solution

$x = \frac{x^{3}}{x^{2}} = \frac{2 + 11 i}{3 + 4 i} = \frac{2 + 11 i}{3 + 4 i} \times \frac{3 - 4 i}{3 - 4 i} = \frac{6 - 8 i + 33 i - 44 i^{2}}{9 - 16 i^{2}} = \frac{50 + 25 i}{25} \Rightarrow 2 + i = a + i b ∴ a = 2, b = 1 \Rightarrow a + b = 2 + 1 = 3$

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