Col. Rbtx
03-14-2006, 12:57 PM
Rbtx's Function of X conjectures
by Robt. W. Brown
March 2006
Let's suppose we have a function of X.. that has 2 dependent variables (or coordinates) defined as:
Y= AX^2.................eq.101
Z= AX^2 + BX.............eq.201
Let's suppose we have a second function of x.. that has 2 dependent variables (or coordinates) defined as:
y= -BX...................eq. 301
z= AX^2 + BX..............eq.401
Just by looking at the above functions it becomes obvious that anytime Y = y , a common intercept point will occur. Simply because, Z always equals z and the independent variable X is always the same in both equations:
Z= z
Because:
AX^2 + BX= AX^2 + BX
Next let's make these special conditions about the constants A and B:
1. Neither A or B can ever equal zero.
2. A or B can be any number in the real-number set, except zero.
3. A and B can either be integers or fractions.
4. A and B can even sometimes be equal.
**********
Now returning to the two functions expressed above:
If ...Y= y ...........assumption #1
This would mean:
AX^2 = -BX.......eq.501
We already know:
Z=z
Then according to assumption #1, we might conclude:
**********************************
X=X..........(the independent variable)
Y= y
Z=z
WE would then have to conclude that the conditions for an 'intercept' in 3-dimensional space (and graphing) would have been met, if what I just stated above is true!
Now returning to equation 501:
AX^2 = -BX.......eq.501
Rearranged, equation 501 always can be stated:
AX^2 + Bx = 0 ............eq.601
Then, if we assume that C= 0:
Equation 601 can become a complete quadratic:
AX^2 + BX + C = 0 ..............eq.701
Now we can make a series of conjectures:
1. Anytime two diverse functions of X can be stated in the basic forms that are laid out above and be manipulated into the quadratic form of eq.701 where C=0 and if the Constants A & B also conform to the conditions set-forth above, the following 'conjectures' will most probably always prove true:
A. There will always be 2 common intercept points.
B. There will only be 2 common intercept points in the real number set.
C. One intercept will always occur at origin.
D. Either root of the related quadratic equation will always compute and insure a common intercept point.
E. Since there are always two roots in a quadratic, this cooincides with conjecture A above.
F. If it is found that the conjectures above are generally true in basic geometry
but conflict with other forms of advanced geometry, this will indicate that those branches of geometry are in conflict with quadratic laws.
If you create a problem where these conjectures prove false, equation 701 would have to be untrue or C would have to be greater or less than zero (in your problem). (i.e)
AX^2 + BX + C = 0 ..............eq.701
rearranged:
AX^2 = -BX + C ......
If C is greater or less than zero, then:
Y =/= y
Because:
Y = AX^2
y= -BX
So when:
Y= y
by subbing, this becomes:
AX^2 = -BX
Proving that C must equal zero, at least to insure an intersection of the two diverse functions in question, in the manner I have described above.
**
Compliments of Col. Rbtx , the Barnyard Physicist of Texas
*******
by Robt. W. Brown
March 2006
Let's suppose we have a function of X.. that has 2 dependent variables (or coordinates) defined as:
Y= AX^2.................eq.101
Z= AX^2 + BX.............eq.201
Let's suppose we have a second function of x.. that has 2 dependent variables (or coordinates) defined as:
y= -BX...................eq. 301
z= AX^2 + BX..............eq.401
Just by looking at the above functions it becomes obvious that anytime Y = y , a common intercept point will occur. Simply because, Z always equals z and the independent variable X is always the same in both equations:
Z= z
Because:
AX^2 + BX= AX^2 + BX
Next let's make these special conditions about the constants A and B:
1. Neither A or B can ever equal zero.
2. A or B can be any number in the real-number set, except zero.
3. A and B can either be integers or fractions.
4. A and B can even sometimes be equal.
**********
Now returning to the two functions expressed above:
If ...Y= y ...........assumption #1
This would mean:
AX^2 = -BX.......eq.501
We already know:
Z=z
Then according to assumption #1, we might conclude:
**********************************
X=X..........(the independent variable)
Y= y
Z=z
WE would then have to conclude that the conditions for an 'intercept' in 3-dimensional space (and graphing) would have been met, if what I just stated above is true!
Now returning to equation 501:
AX^2 = -BX.......eq.501
Rearranged, equation 501 always can be stated:
AX^2 + Bx = 0 ............eq.601
Then, if we assume that C= 0:
Equation 601 can become a complete quadratic:
AX^2 + BX + C = 0 ..............eq.701
Now we can make a series of conjectures:
1. Anytime two diverse functions of X can be stated in the basic forms that are laid out above and be manipulated into the quadratic form of eq.701 where C=0 and if the Constants A & B also conform to the conditions set-forth above, the following 'conjectures' will most probably always prove true:
A. There will always be 2 common intercept points.
B. There will only be 2 common intercept points in the real number set.
C. One intercept will always occur at origin.
D. Either root of the related quadratic equation will always compute and insure a common intercept point.
E. Since there are always two roots in a quadratic, this cooincides with conjecture A above.
F. If it is found that the conjectures above are generally true in basic geometry
but conflict with other forms of advanced geometry, this will indicate that those branches of geometry are in conflict with quadratic laws.
If you create a problem where these conjectures prove false, equation 701 would have to be untrue or C would have to be greater or less than zero (in your problem). (i.e)
AX^2 + BX + C = 0 ..............eq.701
rearranged:
AX^2 = -BX + C ......
If C is greater or less than zero, then:
Y =/= y
Because:
Y = AX^2
y= -BX
So when:
Y= y
by subbing, this becomes:
AX^2 = -BX
Proving that C must equal zero, at least to insure an intersection of the two diverse functions in question, in the manner I have described above.
**
Compliments of Col. Rbtx , the Barnyard Physicist of Texas
*******