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.

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Is $a+5^{1/3}b+5^{2/3}c$ a root of any cubic polynomial in $\mathbb{Q}$?

For arbitrary $a, b, c \in \mathbb{Q}$, let $w := a + 5^{1/3} b + 5^{2/3} c$, is $w$ a root of any cubic polynomial in $\mathbb{Q}$? I guess the cubic polynomial always exists. But I am confused about ...
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How to find $x$ by $y$ in bezier curve.

I draw the lines connecting the different nodes on the screen with the cubic bezier curve. These lines must be suitable for interaction. So I need to know how close the user is to the line based on ...
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Cubic polynomial without $x^1$

I factorized this equation $4x^3+6x^2-1=0$ I think it's harder to factorize when the $x$ (degree one) doesn't exist here. So, this is what I did now: $2x^2(2x+3)-1=0$ But the factorized equation looks ...
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solve this formula for $x$: $y = \frac{5x(2x^2 + 27x + 89)}{6}$

I need to solve this formula for x: $$y = \frac{5x(2x^2+27x+89)}{6}$$ When inserting this into a Formula calculator it gave me the formula in the attached image, however, I have no clue how to use ...
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Stuck on simplifying expressions involving trig and inverse trig functions

TL; DR Using Mathcad and Wolfram I can see that $$\sqrt{7}\cos\frac{\tan^{-1}\left(\frac{9\sqrt{3}}{10}\right)}{3}=2.5$$ The decimal value seems to be exact because Mathcad displays it like that with ...
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What are the necessary and sufficient conditions for the cubic equation to have at least 1 positive real root?

What are the necessary and sufficient conditions for the cubic equation to have at least 1 positive real root? I'm just dealing with the $2$ simplest cases. Case-1: $$x^3+px+q=0$$ where, $p<0$ and ...
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Condition for real roots of depressed cubic equation

My book sets out to prove that three normals drawn to any parabola $y^2=4ax$ from a given point $(h,k)$ are real when $h>2a$. It gives the following proof: Proof: When normals are real, then all ...
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