Find the real number $x$ if $(x - 2 i) (1 + i)$ is purely imaginary.

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Solution

Given that
$(x - 2 i) (1 + i)$ is purely imaginary

First, simplify the expression

$(x - 2 i) (1 + i) = x (1 + i) - 2 i (1 + i)$

$\Rightarrow (x - 2 i) (1 + i) = x + x i - 2 i - 2 i^{2}$

$\Rightarrow (x - 2 i) (1 + i) = x + (x - 2) i - 2 i^{2}$

we know that $i^{2} = - 1$

So, $\Rightarrow (x - 2 i) (1 + i) = x + (x - 2) i - 2 (- 1)$

$\Rightarrow (x - 2 i) (1 + i) = x + (x - 2) i + 2$

$\Rightarrow (x - 2 i) (1 + i) = (x + 2) + i (x - 2)$

It is given that the expression is purely imaginary

$\Rightarrow (R e a l p a r t) = 0$

$\Rightarrow R e a l p a r t = (x + 2) = 0$

Therefore $\Rightarrow x = - 2$

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