Cubic Definition and 430 Threads

Let's denote ##\sqrt[3] r =t##. The three expressions above can be written as $$x_1=2t \cos \frac {\phi} 3, x_2=t (-\cos \frac {\phi} 3 -\sin \frac {\phi} 3), x_3=t (-\cos \frac {\phi} 3 +\sin \frac {\phi} 3)$$ The Vieta's formulae for the given equation are $$x_1+x_2+x_3=0$$ $$x_1 x_2+ x_2 x_3...

Proof: Let ## x=\sqrt[3]{18+\sqrt{325}}+\sqrt[3]{18-\sqrt{325}} ##. Then ## x^3=(\sqrt[3]{18+\sqrt{325}}+\sqrt[3]{18-\sqrt{325}})^3 ##. Note that ## (a+b)^3=a^3+3ab(a+b)+b^3 ## where ## a=\sqrt[3]{18+\sqrt{325}} ## and ## b=\sqrt[3]{18-\sqrt{325}} ##. This gives ##...

My attempt at a solution: We have three points, hence we will have two polynomials ##p_1(x)## for ##x\in [-5,-4]## and ##p_2(x)## for ##x\in [-4,-3]##. Define: ##p_1(x)=c_1x^3+c_2x^2+c_3x+c_4## and ##p_2(x)=c_5x^3+c_6x^2+c_7x+c_8## where the ##c_n## stuff are coefficients for the polynomial. I...

The problem and solution are posted... no. 8 I may need insight on common difference ... In my lines i have, Let the roots be ##(b), (b-1)## and ##(b+1)##. Then, ##x^3-3bx^2+3cx-d = a(x-b(x-b+1)(x-b-1)## ##x^3-3bx^2+3cx-d= a(x^3-3bx^2+3b^2x-x-b^3+b)## ##a=1##. Let...

(i) I take the second derivative of Y: Y'' = 6X + 2A. Y'' = 0 when X = -A/3. Moreover, as Y'' is linear it changes sign at this X. Thus, it is the point of inflection. (iii) After the substitution, the term x^2 appears twice: one, from X^3 as -3(x^2)(A/3), and another from AX^2 as Ax^2. They...

This is part of a longer exercise I struggled with. I checked the solutions manual, and there was a bit where they performed the following steps: $$x^3=3a^2x-2a^3 \\$$ $$(x-a)^2(x+2a)=0$$ And then concluded that the roots were ##a## and ##-2a##, which is clear. What I can't work out is how...

If I have a air tank that is 1 cubic feet with the pressure gauge at 0 and I pump in air to 5000 psi how many cubic feet of a 100 psi would be in that tank? Any help would be appreciated

In terms of tetrahedral stacking, as occurs in diamond cubic, what I'm describing would be a system with additional tetrahedra in the empty cubes of that figure, but with their vertices on the opposite corners of the cubes that contain them to the "regular" diamond cubic arrangement. Due to the...

Like this one: ##f(x)=3x^3 -18x^2 +36x##.Is there any way to find the inverse of this function without using the general solution to cubic equation?

For a state to be stationary it must be time independent. Naively, I tried to find the values of c where I don't have any time dependency. ##e^{c \cdot L_z} \psi (r,t) = e^{c L_z} \sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}##...

The context: I created an educational resource, a set of interactive diagrams that allow the user to see how Hamilton's stationary action arrrives at the true trajectory. There is a diagram for each of the following three cases: - Uniform force, hence the potential increases linear with...

Desmos.com is a great online graphing utility which I'm sure is familiar to many PF users. I wanted to experiment with the Newton-Raphson method using it so chose solution of cubic graphs as an example. The graph shows a variable cubic on which all turning points and intercepts are calculated...

What is the difference for example between cubic boron phosphide and monolayer boron phosphide?

Why is a third degree polynomial called a cubic polynomial? I just don’t see the connection between 3 and a cube.

Hi, I was trying to find roots of the following cubic polynomial and there are only two roots. I believe there should be three roots. Could you please guide me why there are only two roots? If you say that the "1" repeats itself as a root then I'd say the same could be said of "0.9". Thank...

Find all triples $(a,\,b,\,c)$ of positive integers such that $a^3+b^3+c^3=(abc)^2$.

If $\alpha,\,\beta,\,\gamma$ are the roots of the equation $x^3+ax+1=0$, where $a$ is a positive real number and $\dfrac{\alpha}{\beta},\,\dfrac{\beta}{\gamma},\,\dfrac{\gamma}{\alpha}$ be the roots of the equation $x^3+bx^2+cx-1=0$, find the minimum value of $\dfrac{|b|+|c|}{a}$.

Find a polynomial of degree 3 with real coefficients such that each of its roots is equal to the square of one root of the polynomial $P(x)=x^3+9x^2+9x+9$.

Hey! 😊 Show that the interpolation exercise for cubic splines with $s(x_0), s(x_1), , \ldots , s(x_m)$ at the points $x_0<x_1<\ldots <x_m$, together with one of $s'(x_0)$ or $s''(x_0)$ and $s'(x_m)$ or $s''(x_m)$ has exactly one solution. Could you give me a hint how we could show that? Do...

##\sum ∝=3## ##\sum ∝β=0## ##∝βγ=-4## ##\sum2 ∝=6## ##\sum 2∝.2β##=4##\sum ∝β=0## ##2∝.2β.2γ=-32## we then end up with ##x^3-6x^2+0x+32=0## ##x^3-6x^2+32=0## i am looking for alternative methods ...

An equation $x^3+ax^2+bx+c=0$ has three (but not necessarily distinct) real roots $t,\,u,\,v$. For what values of $a,\,b,\,c$ are the numbers $t^3,\,u^3,\,v^3$ roots of an equation $x^3+a^3x^2+b^3x+c^3=0$?

The roots $x_1,\,x_2$ and $x_3$ of the equation $x^3+ax+a=0$ where $a$ is a non-zero real number, satisfy $\dfrac{x_1^2}{x_2}+\dfrac{x_2^2}{x_3}+\dfrac{x_3^2}{x_1}=-8$. Find $x_1,\,x_2$ and $x_3$.

Prove that there are no integers $a,\,b,\,c$ and $d$ such that the polynomial $ax^3+bx^2+cx+d$ equals 1 at $x=19$ and 2 at $x=62$.

Hi, I was trying to solve the equation x³-2x-5=0. I solved it online and the real solution is x≈2.0946. The following is an excerpt from the Wikipedia article on cubic equation. The cubic equation, x³-2x-5=0, is a depressed cubic and in this case 4p³+27q²>0. But the real root of equation...

Suppose the system of equations (coming from invariance of the wave equation) : $$B=-vE\\A^2-B^2/c^2=1\\E^2-c^2D^2=1\\AD=EB/c^2\\B=vA\\AE-BD=1$$ If one adds a lightspeed movement like $$A=a+f\\B=b-cf\\D=d+h\\E=e-ch$$ Then solving equ 1 for f gives $$f=(b+ve-vch)/c$$ Equ 4 for h implies...

Hi, Can someone please help me in understanding few parameters of cubic equation formula for solving polynomial of degree three. I attached the formula in the screenshot. My questions are: (1) what is ". " dot in the end of the formula and what does it mean? (2) I want to use it only for real...

Prove that $3^n\ge(n+3)^3$ for any natural number $n\ge6$.

I created an array, where the first three entries of each column are the x,y, and z coordinates. The last entry in each column is the charge. I called this array PCQ. l/2 l/2 -l/2 -l/2 -l/2 l/2 l/2 -l/2 -l/2 l/2 l/2 -l/2 -l/2 -l/2 l/2 l/2 l/2 l/2 l/2 l/2 -l/2 -l/2 -l/2 -l/2 q -q q -q q...

Hi PF! Here equations 4 and 5 imply $$a^3=a_0^3+3 a_0 \Sigma \implies a_0 \approx a\left( 1- \frac{1}{a^2}\Sigma \right).$$ Can someone explain how this approximation is found? Edit: I place in calculus thread because there must be some series expansion going on or a neglecting of terms.

##x^3y+4x-32=0## is there a particular method for solving this? i know that ##x=2## and ##y=3##

Summary: Cubic equation How to solve for t ?

I understand how to show a given graph does/does not contain a 1-factor but I'm not sure how to show existence (or the lack thereof). Please advise.

So I am currently looking to work on some projects with my brother. I have some questions regarding the usage of nitrogen. We are looking at a 7.2 cubic metre cylinder of Nitrogen Pure compressed and wondering at what rate of pressure will equal roughly 30-60mins of use? Nitrogen Bottle here...

The equation ## x^3-x+.1=0 ##, has 3 real roots that can be quickly approximated as follows: Writing the equation ## x=.1+x^3 ##, iterative methods quickly indicate that there is a root ## r_1 ## near ## x=+.1 ##, and more accurately ## r_1 \approx + .101 ##. ## \\## Doing an approximate...

Answer is given, but no explanation or logic for it. From HiSet free practice test

If i shon a red laser across the surface of the osmium cube 5mm above the solid perfect 1000mm cube, by how many degrees would it be deflected?

I don't understand this. a is not suppose to be -1; this is the only rule in the equation The answer is the second picture, I just don't know the steps that lead to that answer.

Hi, this question was in a year 11 extension maths textbook in the enrichment section. I have the answer as k>17 and k<-11 because I graphed it on GeoGebra. The Graph can be found here: https://ggbm.at/xpegwwtq. While I know the answers I would like to know how to work it out using algebra...

Hey! :o Let $S_{X,3}$ be the vector space of cubic spline functions on $[-1,1]$ in respect to the points $$X=\left \{x_0=-1, x_1=-\frac{1}{2}, x_2=0, x_3=\frac{1}{2}, x_4\right \}$$ I want to check if the function $$f(x)=\left ||x|^3-\left |x+\frac{1}{3}\right |^3\right |$$ is in $S_{X,3}$...

Hey! :o I want to determine an approximation of a cubic polynomial that has at the points $$x_0=-2, \ x_1=-1, \ x_2=0 , \ x_3=3, \ x_4=3.5$$ the values $$y_0=-33, \ y_1=-20, \ y_2=-20.1, \ y_3=-4.3 , \ y_4=32.5$$ using the least squares method. So we are looking for a cubic polynomial $p(x)$...

Homework Statement We were given a tutorial to complete which I did complete. Now the question is: By modifying the appropriate lines in your script file, find the values of a, b, c, and d so that the cubic polynomial  y = ax3 + bx2 + cx + d  passes through the (x, y) pairs (-1, 3), (0, 8)...

Hello, In general, any equation is a statement of equality between two expressions. In the 2D case, equations generally involve the two variables ##x## and ##y## or either variable alone if we require the other variable to be equal to zero. The most general quadratic equation should be the...

Homework Statement Hi, this is more of a review question and I'm just looking at solutions of natural cubic spline equations and some will give the cubic spline as: 1. s(x) = a + bx + cx^2 + dx^3 on Wolfram while other pages will give: 2. s(x) = a + b(x - t) + c(x-t)^2 + d(x-t)^3 where...

I'm having hard time finding the center of inversion of cubic diamond structure. At first I thought (2,2,2) would be the center of inversion, but (1,1,3), (3,3,3), (1,3,1), (3,1,1) (i.e. four atoms inside the cube) are not centrosymmetric about (2,2,2).

and survive for 24 hours. That would be 4.16 cubic millimeters per mosquito! These findings are due to testing of packing methods for shipping mosquitos around for disease prevention (like malaria which is mosquito transmitted.

I have a tank that is 250 Cubic Feet. The tank is under vacuum (1x10-5 Torr).. which is actually 1.93368e-7 psia. How much Argon, in cubic feet, does it take to back-fill to atmosphere or 14.7 psia? Ambient temperature in tank is 90 degrees.

Homework Statement -x^3 - 6x^2 -12x -8 Homework EquationsThe Attempt at a Solution I don't know, I just know the roots are -2 with multiplicity 3.

f(x) = ax^3 + bx^2 + cx + d min(1, -4) max(-2, 1) find a,b,c,d

In case of simple cubic lattice relative magnetization is given by \sigma=1-\frac{1}{S}\frac{v}{(2\pi)^3}\int^{\frac{\pi}{a}}_{-\frac{\pi}{a}}\int^{\frac{\pi}{a}}_{-\frac{\pi}{a}}\int^{\frac{\pi}{a}}_{-\frac{\pi}{a}}\mbox{d} k_x\mbox{d} k_y\mbox{d}k_z(\mbox{e}^{\frac{E(\vec{k})}{kT}}-1)^{-1}...