Let $z_{1},z_{2} \in C$ be the distinct solutions of the equation $z^{2} + 4z - (1 + 12i) = 0$. Then

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Let $z_{1},z_{2} \in C$ be the distinct solutions of the equation $z^{2} + 4z - (1 + 12i) = 0$. Then $\left| z_{1} \right|^{2} + \left| z_{2} \right|^{2}$ is equal to :

Accepted Answer

Sol. [Key Concept: Solve $z^{2} + 4z - (1 + 12i) = 0$ for the two roots, then add their squared moduli ${|z}_{1}|^{2} + {|z}_{2}|^{2}$.]$${Z^{2} + 4Z - (1 + 12i) = 0 }{\Rightarrow Z = \frac{- 4 \pm \sqrt{16 + 4(1 + 12i)}}{2} }{\Rightarrow Z = - 2 \pm \sqrt{5 + 12i} }{\Rightarrow Z = - 2 \pm (3 + 2i) }{\Rightarrow Z = 1 + 2i, - 5 - 2i }{	herefore\left| Z_{1} ight|^{2} + \left| Z_{2} ight|^{2} }{= 5 + 29 }{= 34 }$$

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