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Is this an identity? ∛(a + √b) - ∛(a + √b) = ∛(a - √b) - ∛(a - √b)?
Does this always hold true? These solutions arise when solving cubics by Cardano's formula, and it seems the ± sign in front of the square roots are irrelevant.
∛(a + √b) - ∛(a + √b) = ∛(a - √b) - ∛(a - √b)
for example:
2x³ - 5x² - 7x - 20 = 0
with Cardano's method would yield:
x = 5/6 +
∛[190/27±√(-67³/36³+190²/27²)] - ∛[-190/27±√(-67³/36³+190²/27²)]
and the both of positive/negative yield the same result.
CORRECTION:
∛(a + √b) - ∛(- a + √b) = ∛(a - √b) - ∛(- a - √b)
2 Answers
- JeremyLv 57 years ago
It was easier than I thought... Taking the negative out of the second term on each side:
we have:
∛(a + √b) + ∛(a - √b) = ∛(a - √b) + ∛(a + √b)
- Francis KLv 57 years ago
Are you sure you have this typed in correctly.
It looks like each side is always equal to zero the way you have it typed.
