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The cubic formula in simplest form
To solve the cubic equation
(1) 3 2
y py qy r + + += 0
we must first remove the quadratic term. This is always achieved with the substitution
(2) 3
p y x = − .
Substituting (2) into (1) we get
(3) 3 x cx a − −= 3 20 ,
where
(4)
2
9 3
p q c = − and
3
6 27 2
pq p r a = −− .
(Equation (3) is known as the reduced cubic.) Now we can write our cubic formula for
the real root of (3).