The solution to a cubic equation (click here for a full
explanation) has the following form:
______
³√x + Ö
y - |
_______
³Ö
-x
+ Ö
y |
All the solutions produced by the cubic formula
have this form, even if the solution is an integer. For
example, |
_________
³√10 + Ö
108 - |
__________
³Ö -10
+ Ö
108 |
= 2 |
I used trial and error to find dozens of integer-valued expressions
of this form, and found they all have the following general form:
|
_________________
³√3rp - 8r³ + pÖ
p-3r² - |
__________________
³Ö -3rp + 8r³ + pÖ
p-3r² |
= 2r |
| where r is any real number, and p³3r² |
|
Now I will prove this in two cases, Case I, where p=3r², and
Case II, where p>3r².
CASE I, when p=3r²
|
_______
³√3rp - 8r³ - |
________
³Ö -3rp+ 8r³ |
= |
__________
³√3r(3r²)- 8r³ - |
__________
³Ö -3r(3r²)+8r³ |
= |
_______
³√9r³- 8r³ - |
________
³Ö -9r³+8r³ |
= |
__
³√r³ - |
___
³Ö -r³ |
= r-(-r) = 2r |
let 2w = |
_________________
³√3rp - 8r³ + pÖ
p-3r² - |
__________________
³Ö -3rp + 8r³ + pÖ
p-3r² |
|
For simplicity in manipulating this equation,
let x = 3rp - 8r³, and
In this proof we will use the following facts about x and y:
x2 = 64r6 - 48r2p + 9r2p2,
and
y2 = -3r2p2 + p3.
Subtracting one from the other, we get
x2 - y2 = 64r6 - 48r2p
+ 12r2p2 - p3
x2 - y2 = (4r2 - p)3
Substituting x and y in our original equation, we now have, |
2w = |
____
³√x + y - |
_____
³Ö -x + y |
|
8w³ = ( |
____
³√x + y - |
_____
³Ö -x + y |
)³ |
8w
³ = (x+y) - 3 |
__________
³√(x+y)²(-x+y) + 3 |
__________
³√(x+y)(-x+y)² - |
(-x+y) |
8w
³ = 2x - 3 |
___________
³√(-x²+y²)(x+y) + 3 |
____________
³√(-x²+y²)(-x+y) |
|
8w
³ = 2x - 3 |
_______
³√(-x²+y²) ( |
____
³√(x+y) - |
_____
³√(-x+y) |
) |
8w³ = 2x - 3(p-4r²) (2w)
4w³ = x - 3(p-4r²) (w)
4w³ = 3rp - 8r
³ - 3(p-4r²) (w)
4w³ = 3rp - 8r
³ - 3wp + 12r²w
4w³ = 3rp - 3wp - 8r
³ + 12r²w
0 = 3rp - 3wp - 8r³ + 12r²w - 4w
³
0 = 3rp - 3wp - (8r³ - 12r²w - 4rw² + 4rw²+
4w³)
0 = 3rp - 3wp - (8r³ - 4r²w - 4rw²) - (-8r²w + 4rw²
+ 4w³)
0 = 3rp - 3wp - r(8r² - 4rw - 4w²) + w(8r² - 4rw - 4w²)
0 = (r-w)(3p) - (r-w)(8r² - 4rw - 4w²)
0 = (r-w)(3p - (8r² - 4rw - 4w²))
Now we can see that either r=w (which is the thing we're trying
to prove) or else
3p = 8r² - 4rw - 4w². So if 3p is not equal to 8r²
- 4rw - 4w², then r=w, and the proof
is complete.
(r+2w)² >= 0
r² + 4rw + 4w² >= 0
r² >= - 4rw - 4w²
9r² >= 8r² - 4rw - 4w²
Remember that p>3r², from the assumption of Case II,
above, so 3p>9r².
3p > 9r² >= 8r² - 4rw - 4w²
3p > 8r² - 4rw - 4w²
So 3p is not equal to 8r² - 4rw - 4w²,
proving that r=w, or in other words,
|
_________________
³√3rp - 8r³ + pÖ
p-3r² - |
__________________
³Ö -3rp + 8r³ + pÖ
p-3r² |
= 2r |
| where r is any real number, and p³3r² |
|
Related pages in this website
Quadratic formula
|
