Locating the roots of a quadratic equation in one variable through a Line-Circumference (LC) geometric construction in the plane of complex numbers
Bibliographic record
Abstract
This paper describes a geometrical method for finding the roots of a quadratic equation in one complex variable, by means of a Line L and a Circumference C in the complex plane, constructed from the known coefficients of the quadratic equation. This Line-Circumference (LC) geometric structure contains the sought roots at the intersections of its component elements L and C. Line L is mapped onto Circumference C by a Möbius transformation. The location and inclination angle of L can be computed directly from the coefficients, while C is constructed by dividing the constant term of the quadratic equation by each point from L. This paper describes the technical details for the quadratic LC Method, and then shows how the quadratic LC Method works through a numerical example. The quadratic LC method described here can be extended to find initial approximations to the roots of polynomials in one variable of degree greater than two. As an additional feature, this paper also studies an interesting property of the rectilinear segments connecting key points in a quadratic LC structure.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".