Show that the roots of the equation $x^{2} + 2 (a + b) x + 2 (a^{2} + b^{2}) = 0$ are unreal.

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Solution

For unreal roots $D < 0$
$b^{2} - 4 a c < 0$
$B = 2 (a + b)$ $C = 2 (a^{2} + b^{2})$
$B^{2} - 4 A C < 0$ $A = 1$
$4 (a + b)^{2} - 4 \times 1 \times 2 (a^{2} + b^{2})$
$4 a^{2} + 4 b^{2} + 8 a b - 8 a^{2} - 8 b^{2}$
$8 a b - 4 a^{2} - 4 b^{2}$
$= - 4 (a - b)^{2} < 0$
Hence roots are unreal.

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