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

Let α be the real root and iβ be the imaginary root of the given equation. Thenα+iβ=−a⇒ α−iβ=−a¯So, 2α=−(a+a¯) and 2iβ=−(a−a¯)Multiplying these, we get4iαβ=a2−(a¯)2∴ 4b=a2−(a¯)2