Quadratic inequalities for complex variables?

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Discussion Overview

The discussion revolves around the properties of quadratic inequalities in the context of complex variables, particularly focusing on the implications of having no real roots for a quadratic polynomial when evaluated at real numbers.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions the assertion that a real parameter λ has no real roots, suggesting this implies λ must be imaginary or complex, which seems contradictory.
  • Another participant clarifies that λ is an arbitrary real number and that the quadratic polynomial P(z) has roots in the complex plane, not λ itself.
  • A participant proposes that if the quadratic polynomial P(λ) is greater than zero for all real λ, then it must have no real roots.
  • There is a discussion about the nature of the roots, with one participant suggesting that the roots must be purely imaginary or complex, not purely real.
  • Another participant confirms that a quadratic with real coefficients has no real roots if the discriminant is negative, linking this to the requirement for the discriminant to be less than zero.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the quadratic inequality and the nature of the roots, indicating that multiple competing views remain without a clear consensus.

Contextual Notes

The discussion includes assumptions about the nature of the roots based on the discriminant and the properties of quadratic functions, which may not be universally applicable without further clarification.

mathsciguy
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Hello, I was looking at Riley's solution manual for this specific problem. Along the way, he ended up with a quadratic inequality:

View attachment how.bmp

If you looked at the image, he said it is given that λ is real, but he asserted that λ has no real roots because of the inequality. Doesn't that mean λ is imaginary or complex in some points then, contradicting his first statement? I reckon this has something to do with the properties of a quadratic inequality for complex variables.
 
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mathsciguy said:
Hello, I was looking at Riley's solution manual for this specific problem. Along the way, he ended up with a quadratic inequality:

View attachment 61729

If you looked at the image, he said it is given that λ is real, but he asserted that λ has no real roots because of the inequality. Doesn't that mean λ is imaginary or complex in some points then, contradicting his first statement? I reckon this has something to do with the properties of a quadratic inequality for complex variables.

\lambda doesn't have roots; it's an arbitrary real number. The quadratic P: z \mapsto az^2 + bz + c has roots, which are those z \in \mathbb{C} for which P(z) = 0. The point is that if P(\lambda) > 0 for all real \lambda then P has no real roots, because if z is real then P(z) \neq 0 and z cannot be a root of P.
 
That's cool, I get it now. Then that means the roots are either purely imaginary or complex (but not purely real) right? Then why is it required that the discriminant be less than zero? Is it because it will make sure that part of the solution will have an imaginary part?
 
Last edited:
mathsciguy said:
That's cool, I get it now. Then that means the roots are either purely imaginary or complex (but not purely real) right? Then why is it required that the discriminant be less than zero? Is it because it will make sure that part of the solution will have an imaginary part?

Yes. A quadratic with real coefficients has no real roots if and only if the discriminant is negative.
 

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