If $x = 1 + 2 i$ then prove that $x^{3} + 7 x^{2} - 13 x + 16 = - 29.$

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Solution

$x = 1 + 2 i$
Converting above equation quadratic form,
$⟹ x - 1 = 2 i ⟹ {(x - 1)}^{2} = {(2 i)}^{2} ⟹ x^{2} + 1 - 2 x = 4 i^{2} ⟹ x^{2} + 1 - 2 x = - 4 ⟹ x^{2} - 2 x + 1 + 4 = 0 ∴ x^{2} - 2 x + 5 = 0$
Given equation, $x^{3} + 7 x^{2} - 13 x + 16 = - 29$
$= x^{3} + 9 x^{2} - 2 x^{2} + 5 x - 18 x + 45 - 29 = x (x^{2} - 2 x + 5) + 9 x^{2} - 18 x + 45 - 29 = x (0) + 9 (x^{2} - 2 x + 5) - 29 = x (0) + 9 (0) - 29 = - 29 = R H S [P r o v e d]$

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