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Question Number 97936 by bobhans last updated on 10/Jun/20

Find the value of (√(45−(√(2000)) )) + (√(45+(√(2000))))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\sqrt{\mathrm{45}−\sqrt{\mathrm{2000}}\:}\:\:+\:\:\sqrt{\mathrm{45}+\sqrt{\mathrm{2000}}}\: \\ $$

Commented by pranesh last updated on 10/Jun/20

find y

$${find}\:{y} \\ $$

Answered by john santu last updated on 10/Jun/20

let z_1 = (√(45−(√(2000)))) , z_2 = (√(45+(√(2000)))) ⇔z_1 ^2 + z_2 ^2 = 90 ∧ z_1 z_2 (√((45−(√(2000)))(45+(√(2000))))) = 5 then (z_1 +z_2 )^2 = z_1 ^2 + z_2 ^2 + 2z_1 z_2 = 100 as both z_1 and z_2 are positive we have z_1 + z_2 = 10

$$\mathrm{let}\:\mathrm{z}_{\mathrm{1}} \:=\:\sqrt{\mathrm{45}−\sqrt{\mathrm{2000}}}\:\:,\:\mathrm{z}_{\mathrm{2}} \:=\:\sqrt{\mathrm{45}+\sqrt{\mathrm{2000}}} \\ $$$$\Leftrightarrow\mathrm{z}_{\mathrm{1}} ^{\mathrm{2}} \:+\:\mathrm{z}_{\mathrm{2}} ^{\mathrm{2}} \:=\:\mathrm{90}\:\wedge\:\mathrm{z}_{\mathrm{1}} \mathrm{z}_{\mathrm{2}} \:\sqrt{\left(\mathrm{45}−\sqrt{\mathrm{2000}}\right)\left(\mathrm{45}+\sqrt{\mathrm{2000}}\right)}\:=\:\mathrm{5} \\ $$$$\mathrm{then}\:\left(\mathrm{z}_{\mathrm{1}} +\mathrm{z}_{\mathrm{2}} \right)^{\mathrm{2}} \:=\:\mathrm{z}_{\mathrm{1}} ^{\mathrm{2}} \:+\:\mathrm{z}_{\mathrm{2}} ^{\mathrm{2}} \:+\:\mathrm{2z}_{\mathrm{1}} \mathrm{z}_{\mathrm{2}} \:=\:\mathrm{100} \\ $$$$\mathrm{as}\:\mathrm{both}\:\mathrm{z}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{z}_{\mathrm{2}} \:\mathrm{are}\:\mathrm{positive}\: \\ $$$$\mathrm{we}\:\mathrm{have}\:\mathrm{z}_{\mathrm{1}} \:+\:\mathrm{z}_{\mathrm{2}} \:=\:\mathrm{10}\: \\ $$

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