Prove that the roots of the equation $x^{2} - 2 a x - a^{2} - b^{2} - c^{2} = 0$ are always real, $a, b, c \in R$

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Solution

$x^{2} - 2 a x - (a^{2} + b^{2} + c^{2}) = 0$
Discriminant, $D = (- 2 a)^{2} - 4 (1) (- a^{2} - b^{2} - c^{2}) = 16 a^{2} + 4 b^{2} + 4 c^{2} \geq 0 ⟹$ given equation has real roots

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