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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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 ?
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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 ...
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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 ...
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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
$$\begin{eqnarray*}
-p &=& x_1 + x_2 +...
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What equation produces this curve?
I'm working on an engineering project, and I'd like to be able to input an equation into my CAD software, rather than drawing a spline.
The spline is pretty simple - a gentle curve which begins and ...
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Can $x^3+3x^2+1=0$ be solved using high school methods?
I encountered the following problem in a high-school math text, which I wasn't able to solve using factorization/factor theorem:
Solve $x^3+3x^2+1=0$
Am I missing something here, or is indeed a more ...
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Integer solutions to $x^3=y^3+2y+1$?
Find all integral pairs $(x,y)$ satisfying $$ 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,$$ but I know this is completely helpless. Please help me in ...
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Solving $ax^3+bx^2+cx+d=0$ using a substitution different from Vieta's?
We all know, a general cubic equation is of the form
$$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}$$
We ...
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Why should it be $\sqrt[3]{6+x}=x$?
Find all the real solutions to:
$$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{...
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 ...
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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. ...
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Question Regarding Cardano's Formula
In Cardano's derivation of a root of the cubic polynomial $f(X)=X^3+bX+c$ he splits the variable $X$ into two variables $u$ and $v$ together with the relationship that $u+v=X$. From this he finds that ...
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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 ...
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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 ...
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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 ...
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