Let $k$ be a real parameter, and consider the equation $$x^3 – x = k – k^3 .$$ Obviously, $x=-k$ is a solution. Is it the only one? How to prove it?

Problem Statement:- If the cubic equation $f(x)=0$ has three real roots $\alpha$, $\beta$ and $\gamma$ such that $\alpha\lt\beta\lt\gamma$, show that the equation $$f(x)+2f'(x)+f”(x)=0$$ has a real root between $\alpha$ and $\gamma$. Attempt at a solution:- If $f(x)$ is a quadratic equation whose roots are $\alpha$, $\beta$ and $\gamma$, then $$f(x)=(x-\alpha)(x-\beta)(x-\gamma)$$ From this we get $f'(x)$ […]

While answering this question, I got myself stumped with this crazy system with an evil graph: $$\begin{cases} 18xy^2+x^3=12 \\ 27x^2y+54y^3=38 \end{cases}$$ and I wonder whether there is some slick method to find the only real root $(x, y)=(2, 1/3)$ without relying on Cardano’s formula, ideally giving some intuition. This closely reassembles some kind of elliptic […]

I’m trying to solve the following trig equation: $\cos^3(x)-\sin^3(x)=1$ I set up the substitutions $a=\cos(x)$ and $b=\sin(x)$ and, playing with trig identities, got as far as $a^3+a^2b-b-1=0$, but not sure how to continue. Is there a way to factor this so I can use the zero product property to solve? Thanks for any help/guidance! P

This question A Diophantine equation involving factorial made me try to find a useful structure for the set of positive integers that are the difference of two cubes. I have two questions similar to the linked question : Define $S$ to be the set of positive integers $N$, that are the difference of two positive […]

Let $x^3-(m+n+1)x^2+(m+n-3+mn)x-(m-1)(n-1)=0$, be a cubic polynomial with positive roots, where $m,n \ge2$ are natural nos. For fixed $m+n$, say $15$, it turns out that least root of the polynomial will be smallest in case of $m=2,n=13$ i.e the case in which difference is largest. I checked it for many values of $m+n$ and same thing […]

When solving for roots to a cubic equation, the sign of the $\Delta$ tells us when there will be 3 distinct real roots (as long as the first terms coefficient, $a$, is non-zero.) Namely when $\Delta$ is positive. The equations to find the 3 roots are: $x_1 = -\frac{1}{3a}(b + C + \frac{\Delta_0}{C})$ $x_2 = […]

I am wondering how to derive the following simplification without knowing it beforehand: $$^3\sqrt{10 + 6\sqrt{3}} = 1 + \sqrt{3}$$ After the fact, it is easy to verify algebraically. The problem arose when applying Cardano’s method to solve $$y^3 + 6y = 20$$ I was able to derive a similar but less complicated simplification, $$\sqrt{3 […]

Given the rational Diophantine equation, $$t^3 – t^2 – \tfrac{1}{3}(n^2 + n)t – \tfrac{1}{27}n^3=w^3\tag1$$ Two points are, $$t_0 = 0\tag2$$ $$t_2 = \frac{-(1 + 2 n) (1 + 11 n + 42 n^2 + 14 n^3 + 13 n^4)}{9 (7 + 14 n + 24 n^2 + 17 n^3 + 19 n^4)}\tag3$$ Question: How do […]

With $\mathrm {gcd}(x,y)=1$ I have the following equation: $$x^3-xy^2+1=N$$ I want to find the integer solutions, given an N, of the variables $x$ and $y$. I have tried factoring the equation into $x(x^2-y^2)=N-1$ and then tried to link it to Pell’s Equation but so far I’ve got nothing, I don’t even understand how Pell’s Equation […]

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