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Let origin and the non- real roots of 2z2+2z+λ=0 form the three vertices of an equilateral triangle in the Argand plane then 3λ=


Solution

2z2+2z+λ=0
Let the roots be z1,z2
z1+z2=1,z1z2=λ2

0,z1,z2 form an equilateral triangle .
z1=z2eiπ/3 (using rotation)
1z2=z2ω2  [eiπ/3=ω2]
z2=11ω2

λ=2z2(ω2)z2=2ω2z22=2ω21+ω2ω2=23, [ω3=1,1+ω+ω2=0]3λ=2

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