First slide

Algebra of complex numbers

Question

If a and b are complex and one of the roots of the equation x² + ax + b = 0 is purely real whereas the other is purely imaginary, then

Moderate

A

$a^{2} - (\bar{a})^{2} = 4 b$
B

$a^{2} - (\bar{a})^{2} = 2 b$
C

$b^{2} - (\bar{b})^{2} = 4 a$
D

$b^{2} - (\bar{b})^{2} = 2 a$

Solution

Let $α$ be the real root and i $β$ be the imaginary root of the given equation. Then
$α + iβ = - a$
$\Rightarrow α - iβ = - \bar{a}$
So, $2 α = - (a + \bar{a}) and 2 iβ = - (a - \bar{a})$
Multiplying these, we get
$\begin{aligned} 4 iαβ = a^{2} - (\bar{a})^{2} \\ ∴ 4 b = a^{2} - (\bar{a})^{2} \end{aligned}$

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