[alogo] Solving a cubic graphically

Following [Adler, p.250] the graphical solution of the cubic equation
                                                                     x3 + ax2 + bx + c = 0,
can be reduced to the location of the intersection points of the two conics
                                                                     y = x2,
                                                                     yx + ay + bx + c =0.
The first is always the same well known parabola. The second is a rectangular hyperbola discussed in detail in Rectangular_Hype_From_Line.html .
Its center is at (-a, -b) and its asymptotes are parallel to the axes. The coefficients (a,b) are in reverse order from those in the above reference.
The figure below illustrates the method:
1) First draw the line  bx+ay+c=0 and find its intersection points {A,B} with the axes.
2) Then find their symmetrics {A',B'} with respect to (-a,-b).
3) Find the orthocenter H of one of the triangles formed by the parallelogram ABB'A' and its diagonals (here BAB').
4) Pass a conic throught the five points {A,B,A',B',H}. This coincides with  yx+ay+bx+c=0.
5) Draw the parabola y=x2.
6) Find the intersection point(s)  (sx, sy) of the two curves. Number sx is a root of the original equation.

[0_0] [0_1] [0_2] [0_3] [0_4] [0_5]
[1_0] [1_1] [1_2] [1_3] [1_4] [1_5]
[2_0] [2_1] [2_2] [2_3] [2_4] [2_5]
[3_0] [3_1] [3_2] [3_3] [3_4] [3_5]
[4_0] [4_1] [4_2] [4_3] [4_4] [4_5]
[5_0] [5_1] [5_2] [5_3] [5_4] [5_5]

See Also

Rectangular_Hype_From_Line.html

Bibliography

[Adler] August Adler Theorie der Geometrischen Konstruktionen G. J. Goeschensche Verlagshandlung, Leipzig 1906

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