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.
1,416 questions
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How do we know which root of the decic is the radius of the circle?
I. Three circles
In this post, the OP asked for the radius $r$ of the central circle,
if the radii of the other three are $(d,e,f)=(3,6,7)$, respectively. Heropup gave the answer $r \approx 4.9648$ ...
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On proving $3(2L+9)-B^2-2\frac{8L+27}{B}>0$ [closed]
For this question: my final interest is to inspect the extraneous solutions of a quartic equation derived from a main radical equation to come up with the two roots of the main radical equation.
The ...
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On determining the signs of the roots of a quartic equation $x^4+8(3+L)x^3+24L(L+4)x^2+32L(L^2+3L-2)x+16L^2(L^2-4)=0.$
Referring to my previous 1st and 2nd questions, to solve the following equation:$$\sqrt{1-x}-\sqrt{1-2x}-\frac{x}{2}=L\quad(0<L\le\sqrt{0.5}-0.25=0.4571)$$
I converted it to a quartic equation as ...
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On solving for positive $x$ satisfying $\sqrt{1-x}-\sqrt{1-2x}=\frac{x}{2}+L$ for $0<L<0.2$
Regarding solving for positive $x$ satisfying $$\sqrt{1-x}-\sqrt{1-2x}=\frac{x}{2}+L$$ for $0<L<0.2$, I converted the main equation into a quartic equation:
$$x^4+8(3+L)x^3+24L(L+4)x^2+32L(L^2+...
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Integer solutions of $x^3 + y^3 +xy=25$ [closed]
My teacher couldn't solve this Diophantine equation and I'm looking to solve it somehow maybe you could help me please. It says:
Find all pairs of integers $(x,y)$ that satisfy the equation $$x^3 + y^...
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Is it possible for a cubic function and a linear function to never intersect? [closed]
Is it possible to device a cubic function (i.e. a function of the form $f(x) = ax^3 + bx^2 + cx + d$) and a linear function (i.e. a function of the form $f(x) = px + q$)
that have zero intersections ...
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A cubic curve locus of points in a triangle whose associated cevians cut its sides in a certain balanced way
This question finds its origin in a somewhat mysterious constraint in a recent question.
Here is the context (see Fig. 1).
Fig. 1. Being given a point $M$ inside triangle $ABC$, its associated ...
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Prove that $f(x)=ax^3+bx^2-2bx+4b$ has at most one real root.
To Prove: The function $f(x)=ax^3+bx^2-2bx+4b$ has at most one real root.
My approach:
I know I could use the extrema $x=\frac{-b\pm\sqrt{b^{2}+6ab}}{3a}$ but substituting back into $f(x)$ yields the ...
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General solution for $x^3+px^2+q=0$
There's a fairly well-known formula for cubic equations of the form $x^3+px+q=0$: The transformation $x = z-\frac{p}{3z}$ gives $z^3+q-\frac{p^3}{27}z^{-3}$, and after multiplying by $z^3$, the ...
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Cubic equation for closest point to a function
We have $y = x^2 + 1$ and are trying to find the point closest to $(4, 0)$. For context, this is for a high school calculus class.
Clearly, $l = \sqrt{(x - 4)^2 + ((x^2 + 1) - 0)^2}$, from which we ...
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$f(x)$ is a cubic polynomial $f(k-1)f(k+1)<0$ is NOT true for any integer $k$. Find $f(8)$
$f(x)$ is a monic cubic polynomial and $f(k-1)f(k+1)<0$ is NOT true for any integer $k$.
Also $f'\left(-\frac{1}{4}\right)=-\frac{1}{4}$ and $f'\left(\frac{1}{4}\right)<0$.
Find value of $f(8)$
...
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Not all quadratic extensions over $\mathbb{Q}$ are contained in the compositum of all the splitting fields of irreducible cubics in $\mathbb{Q}[X]$
Question: Let $F$ be the composite of all the splitting fields of irreducible cubics over $\mathbb{Q}$. Prove that
$F$ does not contain all quadratic extensions of $\mathbb{Q}$.
(This is exercise 16 ...
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A doubt in an IIT-JEE-Adv-2025-P1 question
If $a_i,b_i \in \mathbb R$ for $i\in\{1,2,3\}$, define $f:\mathbb R \to \mathbb R, g: \mathbb R\to \mathbb R, h: \mathbb R \to \mathbb R$
$$f(x)=a_1+10x+a_2x^2+a_3x^3+x^4, \quad g(x)=b_1+3x+b_2x^2+...
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Solving system of equations with one affine and two cubic equations
From Hall & Knight's Higher Algebra:
Solve the system of equations $$ \begin{aligned} x^3 + y^3 + z^3 &= 495 \\ x + y + z &= 15 \\ x y z &= 105 \end{aligned} $$
What I tried
We know ...
user1462542
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Differences in implementation of Cardano's method [closed]
Given a cubic
$$ax^3+bx^2+cx+d=0$$
You can divide by $a$, and replace $x$ with $w - \frac{b}{3a}$ to center the cubic and remove the quadratic term
$$
\begin{aligned}
ax^3+bx^2+cx+d &= 0 \\
x^3+\...
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