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Question Number 134461 by pticantor last updated on 04/Mar/21

soit (E) l′equation complex:  z^2 −2pz−q=0  p,q∈C  let u∈C\ u^2 =q  show that if z_1 ,z_2 are solutions of (E),  ∣z_1 ∣+∣z_2 ∣=∣p+u∣+∣p−u∣

$${soit}\:\left(\boldsymbol{{E}}\right)\:{l}'{equation}\:{complex}: \\ $$$$\boldsymbol{{z}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{pz}}−\boldsymbol{{q}}=\mathrm{0} \\ $$$$\boldsymbol{{p}},\boldsymbol{{q}}\in\mathbb{C} \\ $$$$\boldsymbol{{let}}\:\boldsymbol{{u}}\in\mathbb{C}\backslash\:\boldsymbol{{u}}^{\mathrm{2}} =\boldsymbol{{q}} \\ $$$$\boldsymbol{{show}}\:\boldsymbol{{that}}\:\boldsymbol{{if}}\:\boldsymbol{{z}}_{\mathrm{1}} ,\boldsymbol{{z}}_{\mathrm{2}} \boldsymbol{{are}}\:\boldsymbol{{solutions}}\:\boldsymbol{{of}}\:\left(\boldsymbol{{E}}\right), \\ $$$$\mid\boldsymbol{{z}}_{\mathrm{1}} \mid+\mid\boldsymbol{{z}}_{\mathrm{2}} \mid=\mid\boldsymbol{{p}}+\boldsymbol{{u}}\mid+\mid\boldsymbol{{p}}−\boldsymbol{{u}}\mid \\ $$

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