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Let α and β be real numbers, and let z be a complex number. If the quadratic equation z² + αz + β = 0 has two different roots lying on the vertical line Re(z) = 1, which condition must hold?

1. β lies in the interval (−1,0)
2. The modulus of β is 1
3. β is greater than 1
4. β lies in the interval (0,1)

Correct answer: β is greater than 1

Solution

For the quadratic equation to have two distinct roots on the vertical line Re(z) = 1, the roots must be complex conjugates of the form 1 + yi and 1 - yi. This requires that the discriminant is positive, which leads to the condition that β must be greater than 1 to ensure the roots are real and distinct.

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