# 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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### Failing to solve cubic equation

I'm trying to solve a more complex cubic equation but to simplify things as a start I picked this one: $$3\cdot 4^3+2\cdot 4-200=0$$ Here $x$ is $4$. I'm looking at wikipedia and trying to solve ...
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### cubic equation edge cases

Working on general cubic equation solver in form ax^3+bx^2+cx+d=0 And have no clue for special cases: In terms of cubic there should be one real root and two complex, or 3 real roots if coefficients ...
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### How to prove that the real solution to this equation is greater than 1 without solving it? $x^3 - x^2 +2x - 4 =0$ [closed]

I was attempting a problem online and reached a point where I needed to prove that the real solution to this equation is greater than 1 but couldn't seem to find a way to do it. Please advise! If you ...
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### Prime numbers $p$ such that $7p+1$ is a cube [closed]

I am stuck on this assignment. I have to find every prime number $p$ such that $7p+1$ is a cube number. After exploring enough I must say there is no prime $p$ satisfying this condition. I have tried ...
1 vote
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### Why Can't Cubic Equation Have Fractional Solutions When Its Coefficients Are All Integers? [duplicate]

In Leonhard Euler's book, "The Elements of Algebra" he seems to say that if we convert any cubic equation into the form $x^3 + ax^2 + bx + c$, and make sure that $a$, $b$ and $c$ are integer ...
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### Finding rational coefficients of a cubic polynomial that fits 4 data points that have been floored to an integer

I have 4 data points: (204, 5422892) (205, 5722486) (207, 6343357) (213, 8386502) I have information that these data points were generated with a cubic polynomial $y = ax ^ 3 + bx ^ 2 + cx + d$ with ...
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### Can I use this algorithm for solving cubic equations?

I am trying to find the root solutions for a cubic equation including the eigenvalues of each root. I tried to put the equation into my calcualtor but the calculator doesn't show solutions that has ...
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### Convert an expression with radicals into simpler form

It was pointed out in a mathologer video on the cubic formula that $\sqrt{20 + \sqrt{392}} + \sqrt{20 - \sqrt{392}}$ is actually equal to $4$. Is there a series of transformations that can be ...
1 vote
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### Solving a cubic using triple angle for cos (i.e $\cos(3A)$)

a) Show that $x=2\sqrt{2}\cos(A)$ satisfies the cubic equation $x^3 - 6x = -2$ provided that $\cos(3A)$ = $\frac{-1}{2\sqrt{2}}$ I did not have a difficulty with this question, I have provided it for ...
1 vote
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### How do I find a cubic equation given only one root?

Given the root of a cubic equation $Z = \sqrt{Y + \sqrt{Y^2 - \frac{X^6}{27}}} + \sqrt{Y - \sqrt{Y^2 - \frac{X^6}{27}}} - X$ and the assumption that both $X$ and $Y$ are greater than zero, is ...
1 vote
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### Best way to solve $\frac{x^3+3}{x^2+1}>\frac{x^3-3}{x^2-1}$

I was wondering what the best way to solve questions like these are? $$\frac{x^3+3}{x^2+1}>\frac{x^3-3}{x^2-1}$$ I can get the answer, which is $(-\infty,-1)\cup(1,3)$. But I'm not sure if I have ...
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### Is there any faster way to factor $x^3-3x+2$?

$$x^3-3x+2$$ $$x^3-3x+x^2+2-x^2$$ $$x^2-3x+2+x^3-x^2$$ $$(x-2)(x-1)+x^2(x-1)$$ $$(x-1)[x^2+x-2]$$ $$(x-1)(x+2)(x-1)$$ Is there a better, faster way to factor this cubic trinomial?
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### Condition for the existence of positive solution to cubic equation

In a physics textbook I have encountered a cubic equation of the form: $$Ax^3-Bx+C=0$$ The book states that there exists a positive solution $x>0$ to this equation if and only if the following ...
1 vote
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### Find sum of all integral values of $r$ such that all roots of the equation $x^3-(r-1)x^2-11x+4r=0$ are also integers

Find sum of all integral values of $r$ such that all roots of the equation $$x^3-(r-1)x^2-11x+4r=0$$ are also integers. What I could do was $$r=\frac{x^3+x^2-11x}{x^2-4}=x+1+\frac{4-7x}{x^2-4}$$ Since ...
### Find all real numbers $a$ for equation $x^3 + ax^2 + 51x + 2023=0$, has two equal roots.
Problem: Find all real numbers $a$ for which the equation, $x^3 + ax^2 + 51x + 2023=0$, has two equal roots. This problem is from an algebra round of a local high school math competition that has ...