Metamath Proof Explorer


Theorem quartfull

Description: The quartic equation, written out in full. This actually makes a fairly good Metamath stress test. Note that the length of this formula could be shortened significantly if the intermediate expressions were expanded and simplified, but it's not like this theorem will be used anyway. (Contributed by Mario Carneiro, 6-May-2015)

Ref Expression
Hypotheses quartfull.a φ A
quartfull.b φ B
quartfull.c φ C
quartfull.d φ D
quartfull.x φ X
quartfull.t0 φ - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 0
quartfull.m0 φ 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 0
Assertion quartfull φ X 4 + A X 3 + B X 2 + C X + D = 0 X = A 4 - 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 + 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 2 - B 3 8 A 2 2 + C - A B 2 + A 3 8 4 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 X = A 4 - 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 - 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 2 - B 3 8 A 2 2 + C - A B 2 + A 3 8 4 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 X = A 4 + 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 + 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 2 - B 3 8 A 2 2 - C - A B 2 + A 3 8 4 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 X = A 4 + 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 - 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 2 - B 3 8 A 2 2 - C - A B 2 + A 3 8 4 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2

Proof

Step Hyp Ref Expression
1 quartfull.a φ A
2 quartfull.b φ B
3 quartfull.c φ C
4 quartfull.d φ D
5 quartfull.x φ X
6 quartfull.t0 φ - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 0
7 quartfull.m0 φ 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 0
8 eqidd φ A 4 = A 4
9 eqidd φ B 3 8 A 2 = B 3 8 A 2
10 eqidd φ C - A B 2 + A 3 8 = C - A B 2 + A 3 8
11 eqidd φ D C A 4 + A 2 B 16 - 3 256 A 4 = D C A 4 + A 2 B 16 - 3 256 A 4
12 eqidd φ B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 = B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4
13 eqidd φ 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 = 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4
14 eqidd φ 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 = 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3
15 eqidd φ 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 = 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2
16 eqidd φ 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 = 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3
17 eqidd φ - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 = - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3
18 eqidd φ 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 2 - B 3 8 A 2 2 + C - A B 2 + A 3 8 4 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 = 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 2 - B 3 8 A 2 2 + C - A B 2 + A 3 8 4 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2
19 eqidd φ 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 2 - B 3 8 A 2 2 - C - A B 2 + A 3 8 4 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 = 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 2 - B 3 8 A 2 2 - C - A B 2 + A 3 8 4 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2
20 1 2 3 4 5 8 9 10 11 12 13 14 15 16 17 6 7 18 19 quart φ X 4 + A X 3 + B X 2 + C X + D = 0 X = A 4 - 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 + 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 2 - B 3 8 A 2 2 + C - A B 2 + A 3 8 4 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 X = A 4 - 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 - 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 2 - B 3 8 A 2 2 + C - A B 2 + A 3 8 4 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 X = A 4 + 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 + 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 2 - B 3 8 A 2 2 - C - A B 2 + A 3 8 4 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 X = A 4 + 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 - 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2 2 - B 3 8 A 2 2 - C - A B 2 + A 3 8 4 2 B 3 8 A 2 + - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 + B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 - 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 + 2 B 3 8 A 2 3 - 27 C - A B 2 + A 3 8 2 + 72 B 3 8 A 2 D C A 4 + A 2 B 16 - 3 256 A 4 2 4 B 3 8 A 2 2 + 12 D C A 4 + A 2 B 16 - 3 256 A 4 3 2 1 3 3 2