Mathematics

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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Why do cubic equations always have at least one real root, and why was it needed to introduce complex numbers?

I am studying the history of complex numbers, and I don't understand the part on the screenshots. In particular, I don't understand why a cubic always has at least one real root . I don't see why the reason is "since $y^3 − py − q$ is positive for…
Tereza Tizkova
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Cubic equation for closest point to a function

We have $y = x^2 + 1$ and are trying to find the point closest to $(4, 0)$ . For context, this is for a high school calculus class. Clearly, $l = \sqrt{(x - 4)^2 + ((x^2 + 1) - 0)^2}$ , from which we get $l^2 = (x - 4)^2 + ((x^2 + 1) - 0)^2$ and so…
Experimental-Mathematics
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How can we solve this equation $x^3-3x^2-x=\sqrt{2}$

We need to find only the positive root for the equation: $$x^3-3x^2-x=\sqrt{2}$$ I start to think firstly in substitution like taking $x=u+1$ this helps me only in eliminating the quadratic term to obtain the following equation:…
Ramez Hindi
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Prove Ramanujan's formula of nested cubic roots $\sqrt[3]{{a}+b\sqrt[3]{r}}$

Ramanujan found that, for for arbitrary $m$ and $n$ $$\sqrt[3]{(m^2+mn+n^2)\sqrt[3]{(m-n)(m+2n)(2m+n)}+3mn^2+n^3-m^3}\\ =\sqrt[3]{\tfrac {(m-n)(m+2n)^2}9}-\sqrt[3]{\tfrac {(2m+n)(m-n)^2}9}+\sqrt[3]{\tfrac {(m+2n)(2m+n)^2}9} $$ Question: Is there a…
Crescendo
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Show $\int_0^\sqrt[3]2(u^3f(u)+u^7/f(u))du=\ln(1+x)-(\sqrt x-\sqrt2)^2+\frac43$ where $f(u)=\sqrt[3]{27+u^6+\sqrt{27(27+2u^6)}}$ and $x^3-2x^2-4x-2=0$

Inspired by this question , I've come up with an exercise that I hope you will find amusing: Show that $$\begin{eqnarray*}…
Jack D'Aurizio
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How to Simplify Roots Given by Cardano’s Formula for the Cubic Equation $x^3+x^2+x+1=0$

Been trying to figure out how to simplify the first root given by this version of Cardano’s Formula: $$x_1=\sqrt[3]{\dfrac{-\left(\dfrac{2b^{3} -9abc+27a^{2} d}{27a^{3}}\right) +\sqrt{\left(\dfrac{2b^{3} -9abc+27a^{2} d}{27a^{3}}\right)^{2}…
Reuben Miller
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How to separate cubic equations into two conic sections: Deep dive into Omar Khayyam

Lots of people have asked how to use Khayyam's method but I am studying for my dissertation so really need to understand the why . What I really don't understand/ can't find useful proofs for is how he separated cubic equations into two conic…
Bountifull
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Solving $ x^3 - 3x + \sqrt 2 = 0$

Okay, suppose we have a cubic equation like this: $$x^3 - 3x + \sqrt{2} = 0$$ This is a "depressed cubic" so it should be possible to solve directly. According to Wikipedia, the rational root test should be able to suggest whether there might be any…
Petr
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Involution on $2\times 2$ matrices

Show that the map on $2\times 2$ matrices \begin{eqnarray} \left( \begin{matrix} a & b\\ c & d \end{matrix} \right)\overset{\Phi}{\mapsto} \left( \begin{matrix} a & b\\ c & d \end{matrix} \right)\cdot \left( \begin{matrix} 2 (a c - b^2) & a d - b…
orangeskid
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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}, ~D~…
user2661923
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General solution for $x^3+px^2+q=0$

There's a fairly well-known formula for cubic equations of the form $x^3+px+q=0$ : The transformation $x = z-\frac{p}{3z}$ gives $z^3+q-\frac{p^3}{27}z^{-3}$ , and after multiplying by $z^3$ , the equation becomes a quadratic in $z^3$ . $z$ can be…
detroitsigma of geometry dash
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Finding value of $f(4)$ for a cubic polynomial under given derivative and sign conditions

Let $f(x)$ be a monic cubic polynomial (i.e., leading coefficient is $1$ ) such that it satisfies the following conditions: There exists no integer $k$ such that $f(k+1) \cdot f(k-1) < 0$ $f^\prime\left(-\frac{1}{4}\right) = -\frac{1}{4}$…
user483801
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Solving a tricky equation $4x^3-5x^2-5 = 0$

How does one solve $4x^3-5x^2-5 = 0$ ? I've tried the substitution $y = x - \frac{5}{12}$ but then I ended up with this monster of an equation: $864y^3 - 450y -1205 = 0$ . Now I'm stuck. Any help would be greatly appreciated.
jkfthd hiifj
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Solution to depressed cubics

First of all I wanted to clarify that this is my first post here. I was trying to find a solution to the general depressed cubic polynomial and was able to get to the right formula but there are some steps I made which I cannot find a proof of.…
asd
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How can we use Cardano's method to solve a real life problem?

I am making a math project for my school. We can make it on any topic, but should involve some college level math. I have chosen 'Cardano's method' as my topic. I will be showing the method to solve a general cubic equation $$ax^3+bx^2+cx+d=0$$…
Deepani Agarwal
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