Editing Cubic equation (section)

==Geometric interpretation of the roots==

===Three real roots===
[[File:Trigonometric interpretation of a cubic equation with three real roots.svg|thumb|right|150px|For the cubic '''({{EquationNote|1}})''' with three real roots, the roots are the projection on the {{mvar|x}}-axis of the vertices {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} of an [[equilateral triangle]]. The center of the triangle has the same {{mvar|x}}-coordinate as the [[inflection point]].]]

Viète's trigonometric expression of the roots in the three-real-roots case lends itself to a geometric interpretation in terms of a circle.<ref name=Nickalls/><ref>{{Citation
 |first= R. W. D.
 |last= Nickalls
 |date= November 1993
 |title= A new approach to solving the cubic: Cardan's solution revealed
 |url= http://www.nickalls.org/dick/papers/maths/cubic1993.pdf
 |journal= The Mathematical Gazette
 |volume= 77
 |issue= 480
 |pages= 354&ndash;359
 |issn= 0025-5572
 |doi=10.2307/3619777
 |jstor= 3619777 |s2cid= 172730765
 }} See esp. Fig. 2.</ref> When the cubic is written in depressed form '''({{EquationNote|2}})''', {{math|''t''<sup>3</sup> + ''pt'' + ''q'' {{=}} 0}}, as shown above, the solution can be expressed as

<math display="block">t_k=2\sqrt{-\frac{p}{3}}\cos\left(\frac{1}{3}\arccos\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)-k\frac{2\pi}{3}\right) \quad \text{for} \quad k=0,1,2 \,.</math>

Here <math>\arccos\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)</math> is an angle in the unit circle; taking {{math|{{sfrac|1|3}}}} of that angle corresponds to taking a cube root of a complex number; adding {{math|−''k''{{sfrac|2{{pi}}|3}}}} for {{math|''k'' {{=}} 1, 2}} finds the other cube roots; and multiplying the cosines of these resulting angles by <math>2\sqrt{-\frac{p}{3}}</math> corrects for scale.

For the non-depressed case '''({{EquationNote|1}})''' (shown in the accompanying graph), the depressed case as indicated previously is obtained by defining {{mvar|t}} such that {{math|''x'' {{=}} ''t'' − {{sfrac|''b''|3''a''}}}} so {{math|''t'' {{=}} ''x'' + {{sfrac|''b''|3''a''}}}}. Graphically this corresponds to simply shifting the graph horizontally when changing between the variables {{mvar|t}} and {{mvar|x}}, without changing the angle relationships. This shift moves the point of inflection and the centre of the circle onto the {{mvar|y}}-axis. Consequently, the roots of the equation in {{mvar|t}} sum to zero.

===One real root===

====In the Cartesian plane====
[[File:Graphical interpretation of the complex roots of cubic equation.svg|thumb|right|300px|The slope of line RA is twice that of RH. Denoting the complex roots of the cubic as {{math|''g'' ± ''hi''}}, {{math|''g'' {{=}} {{overline|''OM''}}}} (negative here) and {{mvar|h}} = {{math|{{sqrt|tan ''ORH''}}}} = {{math|{{sqrt|slope of line ''RH''}}}} = {{math|{{overline|''BE''}}}} = {{math|{{overline|''DA''}}}}.]]

When the graph of a [[cubic function]] is plotted in the [[Cartesian plane]], if there is only one real root, it is the [[abscissa]] ({{mvar|x}}-coordinate) of the horizontal intercept of the curve (point R on the figure). Further,<ref>{{Citation 
|last=Henriquez 
|first=Garcia 
|title=The graphical interpretation of the complex roots of cubic equations 
|journal=[[American Mathematical Monthly]] 
|volume=42 
|issue=6 
|pages=383–384 
|date=June–July 1935 
|doi=10.2307/2301359 
|jstor=2301359 
}}</ref><ref>{{Citation 
|last=Barr 
|first=C. F.
|title= Discussions: Relating to the Graph of a Cubic Equation Having Complex Roots
|journal=[[American Mathematical Monthly]] 
|volume=25 
|issue=6 
|pages=268–269
|year=1918 
|doi= 10.2307/2972885
|jstor=2972885
}}</ref><ref>{{Citation 
|last1=Irwin 
|first1=Frank
|last2=Wright
|first2=H. N. 
|journal=Annals of Mathematics 
|volume=19 
|issue=2
|pages=152–158 
|year=1917 
|doi= 10.2307/1967772 
|title=Some Properties of Polynomial Curves.
|jstor=1967772
}}</ref> if the complex conjugate roots are written as {{math|''g'' ± ''hi''}}, then the [[real part]] {{mvar|g}} is the abscissa of the tangency point H of the [[tangent line]] to cubic that passes through {{mvar|x}}-intercept R of the cubic (that is the signed length OM, negative on the figure). The [[imaginary part]]s {{mvar|±h}} are the square roots of the tangent of the angle between this tangent line and the horizontal axis.{{clarify|reason=Some hints must be given for the proof of the result, as well as how the line RACB is constructed|date=September 2019}}

====In the complex plane====
With one real and two complex roots, the three roots can be represented as points in the complex plane, as can the two roots of the cubic's derivative. There is an interesting geometrical relationship among all these roots.

The points in the complex plane representing the three roots serve as the vertices of an isosceles triangle. (The triangle is isosceles because one root is on the horizontal (real) axis and the other two roots, being complex conjugates, appear symmetrically above and below the real axis.) [[Marden's theorem]] says that the points representing the roots of the derivative of the cubic are the [[focus (geometry)|foci]] of the [[Steiner inellipse]] of the triangle—the unique ellipse that is tangent to the triangle at the midpoints of its sides. If the angle at the vertex on the real axis is less than {{math|{{sfrac|{{pi}}|3}}}} then the major axis of the ellipse lies on the real axis, as do its foci and hence the roots of the derivative. If that angle is greater than {{math|{{sfrac|{{pi}}|3}}}}, the major axis is vertical and its foci, the roots of the derivative, are complex conjugates. And if that angle is {{math|{{sfrac|{{pi}}|3}}}}, the triangle is equilateral, the Steiner inellipse is simply the triangle's incircle, its foci coincide with each other at the incenter, which lies on the real axis, and hence the derivative has duplicate real roots.