Questions tagged [roots-of-cubics]

For questions related to roots of a cubic equation. All of the roots of the cubic equation can be found by the following means: algebraically, trigonometrically or numerical approximations of the roots.

The solutions of of a cubic equation are called the roots of the cubic function. All of the roots of the cubic equation can be found by the following means:

  • algebraically, that is, they can be expressed by a cubic formula involving the four coefficients, the four basic arithmetic operations and $n$th roots (radicals).
  • trigonometrically
  • numerical approximations of the roots can be found using root-finding algorithms such as Newton's method.
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Algebraically Solve $\left[a + b\sqrt{57}~\right]^3 = 540 + 84\sqrt{57}.$

Unclear how valuable this posting is. It really should be limited to specifying that the goal is to denest one level of the radicals, in an expression like $$\left[c + d\sqrt{D}\right]^{1/3} + \left[c - d\sqrt{D}\right]^{1/3} ~c,d,D \in \Bbb{Z},…
user2661923
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Need help understanding cubic formula derivation by Daniel Rui

I am reading a cubic formula derivation here: http://danielrui.com/papers/cubicPolynomial.pdf It looks fairly straight forward. The author defined: $y = \sqrt[3]{u} − \sqrt[3]v$ so far so good, I suppose there can always be a $u$ and $v$ that will…
some user
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geometric solution to cubic equations

Consider the cubic equation $x^3+d=bx^2$ with $ b,d > 0 $. The question is to give a geometric solution to this equation by interesting two conic sections. In class, our teacher showed us how to construct the geometric solution of functions like …
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Solving a cubic equation in exact terms which is the key to solving the question in picture below

I have arrived at the formula below, I need help with solving this cubic equation for h please.