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Questions tagged [cubics]

This tag is for questions relating to cubic equations, these are polynomials with $~3^{rd}~$ power terms as the highest order terms.

A cubic equation has the form $$ax^3 + bx^2 + cx + d = 0 $$ where $~a,~b,~c,~d~$ are complex numbers and $~a \ne 0~.$

By the Fundamental Theorem of Algebra, cubic equation always has $~3~$ roots, some of which might be equal. All cubic equations have either one real root, or three real roots.

All of the roots of the cubic equation can be found algebraically. The roots can also be found trigonometrically. Alternatively, numerical approximations of the roots can be found using root-finding algorithms such as Newton's method.

Applications:

Cubic equations arise in various other contexts.

  • Marden's theorem states that the foci of the Steiner inellipse of any triangle can be found by using the cubic function whose roots are the coordinates in the complex plane of the triangle's three vertices. The roots of the first derivative of this cubic are the complex coordinates of those foci.

  • The area of a regular heptagon can be expressed in terms of the roots of a cubic. Further, the ratios of the long diagonal to the side, the side to the short diagonal, and the negative of the short diagonal to the long diagonal all satisfy a particular cubic equation. In addition, the ratio of the inradius to the circumradius of a heptagonal triangle is one of the solutions of a cubic equation. The values of trigonometric functions of angles related to $~{\displaystyle 2\pi /7}~$ satisfy cubic equations.

  • Given the cosine (or other trigonometric function) of an arbitrary angle, the cosine of one-third of that angle is one of the roots of a cubic.

  • The solution of the general quartic equation relies on the solution of its resolvent cubic.

  • The eigenvalues of a $~3×3~$ matrix are the roots of a cubic polynomial which is the characteristic polynomial of the matrix.

  • The characteristic equation of a third-order linear difference equation or differential equation is a cubic equation.

  • In analytical chemistry, the Charlot equation, which can be used to find the pH of buffer solutions, can be solved using a cubic equation.

  • In chemical engineering and thermodynamics, cubic equations of state are used to model the PVT (pressure, volume, temperature) behavior of substances.

  • Kinematic equations involving changing rates of acceleration are cubic.

  • The speed of seismic Rayleigh waves is a solution of the Rayleigh wave cubic equation.

References:

https://en.wikipedia.org/wiki/Cubic_function

http://mathworld.wolfram.com/CubicFormula.html

http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-cubicequations-2009-1.pdf

1301 questions
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What is an example of real application of cubic equations?

I didn't yet encounter to a case that need to be solved by cubic equations (degree three) ! May you give me some information about the branches of science or criterion deal with such nature ?
Fereydoon Shekofte
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Motivation of Vieta's transformation

The depressed cubic equation $y^3 +py + q = 0$ can be solved with Vieta's transformation (or Vieta's substitution) $y = z - \frac{p}{3 \cdot z}.$ This reduces the cubic equation to a quadratic equation (in $z^3$). Is there any geometric or algebraic…
Martin Brandenburg
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Cubic formula gives the wrong result (triple checked)

I'd like to solve $ax^3 + bx^2 + cx + d = 0$ using the cubic formula. I coded three versions of this formula, described in three sources: MathWorld, EqWorld, and in the book, "The Unattainable Attempt to Avoid the Casus Irreducibilis for Cubic…
Akim
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Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 + x_3 \\ q &=& x_1x_2 + x_1x_3 + x_2x_3 \\ -r &=&…
Matt Gregory
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What equation produces this curve?

I'm working on an engineering project, and I'd like to be able to input an equation into my CAD software, rather than drawing a spline. The spline is pretty simple - a gentle curve which begins and ends horizontal. Is there a simple equation for…
Giffyguy
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Integer solutions to $x^3=y^3+2y+1$?

Find all integral pairs $(x,y)$ satisfying $$ x^3=y^3+2y+1.$$ My approach: I tried to factorize $x^3-y^3$ as $$(x-y)(x^2 + xy + y^2)=2y+1,$$ but I know this is completely helpless. Please help me in solving this problem.
Identicon
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Can $x^3+3x^2+1=0$ be solved using high school methods?

I encountered the following problem in a high-school math text, which I wasn't able to solve using factorization/factor theorem: Solve $x^3+3x^2+1=0$ Am I missing something here, or is indeed a more advanced method necessary to solve this particular…
yroc
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Solving $ax^3+bx^2+cx+d=0$ using a substitution different from Vieta's?

We all know, a general cubic equation is of the form $$ax^3+bx^2+cx+d=0$$ where $$a\neq0.$$ It can be easily solved with the following simple substitutions: $$x\longmapsto x-\frac{b}{3a}$$ We get, $$x^3+px+q=0$$ where, $p=\frac{3ac-b^2}{3a^2}$…
Learner
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Why should it be $\sqrt[3]{6+x}=x$?

Find all the real solutions to: $$x^3-\sqrt[3]{6+\sqrt[3]{x+6}}=6$$ Can you confirm the following solution? I don't understand line 3. Why should it be $\sqrt[3]{6+x}=x$? Thank you. $$ \begin{align} x^3-\sqrt[3]{6+\sqrt[3]{x+6}} &= 6 \\ x^3 &= 6+…
user548054
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How would you find the exact roots of $y=x^3+x^2-2x-1$?

My friend asked me what the roots of $y=x^3+x^2-2x-1$ was. I didn't really know and when I graphed it, it had no integer solutions. So I asked him what the answer was, and he said that the $3$ roots were $2\cos\left(\frac {2\pi}{7}\right),…
Frank
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Does the limit of the cubic formula approach the quadratic one as the cubic coefficient goes to $0$?

The formula for solving a cubic equation of the form $ax^3+bx^2+cx+d=0$ does not seem to yield the quadratic formula for the limit $\lim _{a \rightarrow 0} \text{(cubic formula)}$. But, if one tries the same thing with the quadratic formula the…
Joel
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Sum of cube roots of complex conjugates

When solving the following cubic equation: $$x^3 - 15x - 4 = 0$$ I got one of the solutions: $$x = \sqrt[3]{2 {\color{red}+} 11i} + \sqrt[3]{2 {\color{red}-} 11i}$$ When I calculated it with a hand calculator, it turned out to be exactly $4$. And…
BarbaraKwarc
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Counting the Number of Real Roots of $y^{3}-3y+1$

Here's my question: How many real roots does the cubic equation $y^3-3y +1$ have? I graphed the function and it crossed the x-axis $3$ times. But my professor doesn't want a graphical explanation. So in that case, I was looking at the Fundamental…
John W. Smith
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What's the explanation for these (infinitely many?) Ramanujan-type identities?

Define the function, $$F(\beta) := \sqrt[3]{\beta+x_1}+\sqrt[3]{\beta+x_2}+\sqrt[3]{\beta+x_3}\tag1$$ where, $$x_1 =2\cos\big(\tfrac{2\pi }{7}\big),\;x_2 =2\cos\big(\tfrac{4\pi }{7}\big),\; x_3 = 2\cos\big(\tfrac{8\pi }{7}\big)$$ or the $x_i$ are…
Tito Piezas III
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Expressing the roots of a cubic as polynomials in one root

All roots of $8x^3-6x+1$ are real. (*) The discriminant of $8x^3-6x+1$ is $5184=72^2$ and so the splitting field of $8x^3-6x+1$ has degree $3$. Therefore, all three roots can be expressed as polynomials in any one given root. Indeed, if $a$ is a…
lhf
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