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If a ,b ,c are nonzero complex numbers o...

If `a ,b ,c` are nonzero complex numbers of equal moduli and satisfy `a z^2+b z+c=0,` hen prove that `(sqrt(5)-1)//2lt=|z|lt=(sqrt(5)+1)//2.`

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`|a| = |b| = |c| = r `
Again ` az^(2) + bz = - c `
` rArr |c| = |-az^(2) - bz| le |a||z^(2)| + |b| |z| `
` rArr r le r |z|^(2) + r |z| `
` rArr |z|^(2) + |z| - 1 ge 0 `
` rArr |z| ge (sqrt5 - 1 )/( 2 ) " " (1)`
Also from ` a z ^(2) = - bz - c,`
`|z|^(2) - |z| - 1 le 0 `
` rArr 0 lt |z| le (sqrt5 + 1 )/(2) " " ` (2)
From (1) and (2) ,
` (sqrt5 - 1 )/(2) le |z| le (sqrt 5 + 1 )/( 2 )`
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