Editing Discriminant (section)

===Conic sections===
A [[conic section]] is a [[plane curve]] defined by an [[implicit equation]] of the form
:<math>ax^2+ 2bxy + cy^2 + 2dx + 2ey + f = 0,</math>
where {{math|''a'', ''b'', ''c'', ''d'', ''e'', ''f''}} are real numbers.

Two [[quadratic form]]s, and thus two discriminants may be associated to a conic section.

The first quadratic form is
:<math>ax^2+ 2bxy + cy^2 + 2dxz + 2eyz + fz^2 = 0.</math>
Its discriminant is the [[determinant]]
:<math>\begin{vmatrix} a & b & d\\b & c & e\\d & e & f \end{vmatrix}. </math>
It is zero if the conic section degenerates into two lines, a double line or a single point.

The second discriminant, which is the only one that is considered in many elementary textbooks, is the discriminant of the homogeneous part of degree two of the equation. It is equal to<ref>{{cite book
|title=Math refresher for scientists and engineers
|first1=John R.
|last1=Fanchi
|publisher=John Wiley and Sons
|year=2006
|isbn=0-471-75715-2
|url=https://books.google.com/books?id=75mAJPcAWT8C&pg=PA45
|at=sec. 3.2, p. 45}}
</ref>

:<math>b^2 - ac,</math>

and determines the shape of the conic section. If this discriminant is negative, the curve either has no real points, or is an [[ellipse]] or a [[circle]], or, if degenerated, is reduced to a single point. If the discriminant is zero, the curve is a [[parabola]], or, if degenerated, a double line or two parallel lines. If the discriminant is positive, the curve is a [[hyperbola]], or, if degenerated, a pair of intersecting lines.