Jesús de la Peña Hernández
3
1 4 ( 1 )
1
x
x
− −
=
−
;
3
1 4 ( 2 )
2
x
x
− −
=
−
In general: x^2 + 4x + 3 = 0
7.14.5 RESOLUTION OF THE COMPLETE EQUATION OF THIRD DEGREE (H. H)
First of all we ́ll recall Fig. 6 (Point 7.14) to show how the 3rd degree equation is behind it.
That figure is now completed with Fig. 1 of present Point 7.14.5
Let ́s get a t expression just dependent of: balls coordinates (0,0) and (l,m) ; a angle (whose
tangent is t); the configuration of billiards table (a,b):
t
b
bt+ty=a− ; m at lt
t
at b bt
= − +
− −
2
2
()( )l−at^3 + m+bt^2 −at+b= 0
This means that the orientation given to the ball in O in order to hit the other one placed at (l,m)
after rebounding orthogonally on both tablesides, is the only real root of the equation just ob-
tained. And that is so because the equation has a positive discriminant, according to drawing
scale.
It is important to insist that lines which receive points (0,0) and (l,m) along the folding opera-
tion, are parallel to their respective tablesides, and distant from them as much as the balls are
distant from said tablesides.
2
Y
I
X
X ́
x O
A
2 x 1
F
t=tgα
z
b
t=
b y
a z
t
+
−
=
a l
m y
t
−
−
=
(l,m)
O(0,0)
Y
(a,y) X
(a,b)^1
(z,b)