Mathematics Stack Exchange
Mathematics Stack Exchange

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

1352 questions
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Is it possible to write $\cos \left( \frac{1}{3}\arccos \frac{37}{64}-\frac{\pi }{3} \right)$ as a radical expression of real number

$\cos \left( \frac{1}{3}\arccos \frac{37}{64}-\frac{\pi }{3} \right)=\frac{{{\left( -37-3\text{i}\sqrt{303} \right)}^{1/3}}+{{\left( -37+3\text{i}\sqrt{303} \right)}^{1/3}}}{2}$, the number inside the cubic root is the complex number, not a real…
Liding Yao
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How to solve $y=2x^3+x$ for $x$

I'm doing something that has throw out: $$y=nx^{2n-1}+x$$ and do make any progress with this problem I needed to make x the subject. Correct me if I'm wrong but this seems impossible to work with for any values other than $n=0,1,2$. $n=2$ gives…
Banbadle
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Any pros and cons of the Cardano's method to solve cubic?

Cardano's method can be used to solve all arbitrary cubic equations. However, I doubt this method would be flawless and I am curious what limitations does the Cardano's method have. So far I only know about the Casus irreducible cases. Thank you.
ihavenoname
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Cubic function range garauntees in 0-1 interval?

Let's say i had a cubic function $y = Ax^3+Bx^2+Cx+D$ and I know that $A,B,C,D$ are all in the range $[0,1]$, but i don't know their value. Is there any way to know any minimum or maximum value of $y$ of this function over the interval of $x$ being…
Alan Wolfe
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Roots of a real cubic equation

I have a cubic equation of the form $$x^3-a^2x-b^2=0.$$ It is given that all roots are real, moreover, only one root is positive and the other two are negative. Let the positive root be $\alpha>0$. Can I express $\alpha$ in terms of $a,b$? In the…
Jack
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Solving Cubic Equation

$$ f(x) = ax^3 + (b - ad)x^2 + (c - bd)x - cd $$ where $a = 18, b = 4, c = 20$ and $d = 12$. What value of x satisfies the equation $f(x) = 0$? $$ f(x)=18x^3- 212x^2-28x-240. $$ i was told to slowly try out all $f(x) = 1/-1, 2/-2 , 3/-3$ until i…
Kerry
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Does the equation $x^{3} - x - p = 0$ always has a solution in $\mathbb{R}$

Does the equation $x^{3} - x - p = 0$ always has a solution in $\mathbb{R}$ for all $p \in \mathbb{R}$. Is there a real solution for $x$ for each real number $p$? I am new to the theory of cubic equations.
Vinod
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Solve for $x$: $6(x-3)^3 + 4(x+9)^2 +8x -5x^2 (x-10) = 0$

Solve for $x$ in: $$6(x-3)^3 + 4(x+9)^2 +8x -5x^2 (x-10) = 0$$ So far I've made it to: $x^3 +242x +162 = 0,$ but now I'm stuck.
Cameron
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Cubic Equation Finding Roots

How to prove that a particular cubic equation has three real and distinct roots without finding its discriminant via calculus method? Please do not use mathematical concepts beyond high school/IIT-JEE. Well and if someone is using the method of…
user220382
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Find cubic equation using known points

I'm writing a javascript game, and are working on some interpolation functions, but I don't have enough mathematical knowledge to work out this problem. I want to use a cubic function for interpolation with this formula: y=ax^3 + bx + c I want to…
LongInt
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Issue with modelling word problems as cubic equations

I am having problems with a specific set of word problems, which are meant to be modeled as cubic equations in order to be solved. I will give some examples to specify where I can't solve it. The first one is: "In a rectangular piece of cardboard…
user28519
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Finding X intercept of a cubic equation?

What is the $x$ intercept of $y=(x-2)(x^2+25) $? To find $x$ intercept:$ 0=(x-2)(x^2+25) $ I tried $ 0=(x-2)(x+5)(x+5)$ in which the $X$ intercepts are $2,-5$ and $-5$. Is this correct?
Helena
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Solving Cubic Equation using Cardano's method: $x^3-4x+1=0$

Substituting $x=u+v$, we get $$3uv=4$$ $$u^3+v^3=-1$$ and it follows that, $$u^3=-\frac{1}{2}+\iota \frac{\sqrt{687}}{18}$$ assume, $z=x+\iota y=-\frac{1}{2}+\iota…
0x13
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Confirming whether in the special case of roots nature cubic equation can be easily solved

Suppose we are given that $a+b+c = A $, $ab+bc+ca = B$ , $abc = C$ , We can observe that if we solve by eliminating two variables out of three(a,b,c) from three equations we would end up getting $x^3 - Ax^2 +Bx -C =0$ a cubic equation which is…
Paracetamol
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Determining the roots of the cubic polynomial

Let's say we have a third-order polynomial given as follows: $P(u)=-u^{3}+a_{1}u^{2}+a_{2}u-a_{3}^{2}$, where $a_{1},a_{2},a_{3}$ are all constants. In the specific problem that I am dealing with it is also supposed that $u=k^{2}.$ So, I need to…
ruudvaan
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