# Questions tagged [quartic-equations]

Equations that can be written in the form $p(x) = 0$ for a univariate polynomial $p$ of degree $4$ or $p(X_1, \ldots, X_r) = 0$ for a multivariate polynomial $p$ of total degree $4$. Questions that use this tag should usually also have the polynomial tag.

161 questions
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### Analytical solution to the crossed ladders problem

I'm working on an analytical solution to the crossed ladders problem. The solution is almost done and already useable (see my answer to Crossed Ladders Problem for details). However I'm left with a ...
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### What kind of Planar Quartic Curve might this be?

I'm trying to smoke out the parameters for a family of curves showing up in a particular optimization problem. I have convinced myself that the solutions always lie on a quartic curve, which is ...
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### Find the relation between $m$ and $n$ such that the following equation has four roots. [closed]

Find the relation between $m$ and $n$ such that the following equation has four roots with $m > 0$. $$x^2 + \left(\dfrac{mx}{m + x}\right)^2 = n$$ Well, I know what the answer is. I just want to ...
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### interesting property of $x^4+2x^3+3x^2+4x+5$

I was recently looking at finding the minimum of the general quartic function $ax^4+bx^3+cx^2+dx+e$, for $a>0$. A closed form expression would, of course, be a huge mess, but it's easy to write ...
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### Reduction Rule Approach Used in Factoring a Quartic

Can someone point me to an article or textbook where I can learn more about the reduction technique used in the third answer titled Rational Root Theorem Solution to the question asked in ...
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### Factoring a general biquartic into two quartics

Let $W_0$, $W_1$, $W_2$, and $W_3$ be known real numbers. I have to solve a biquartic equation: \begin{equation} z^8+W_3z^6+W_2z^4+W_1z^2+W_0=0 \notag \end{equation} Of course I could solve the ...
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### Factorise polynomial with real and complex roots

How would you go about finding the roots of the polynomial: $$x^4 +5x^3+4x^2+6x-4=0.$$ I attempt to form two quadratics e.g. $$(x^2+ax+b)(x^2+cx+d)$$ and then tried to expand, collect like terms, ...
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### Name of the Quartic Surface $z=(x^2-a^2)(y^2-a^2)$

Does the surface defined by the following equation have an specific name? $$f(x,y)=(x^2-a^2)(y^2-a^2)$$ I've searched a lot and found that $z=x^2y^2$ is called Crossed Trough. However I didn't find ...
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### Multidimensional Quartic Equations

I know for the quadratic case (with $A$ an operator): $$ax^2 \Rightarrow x^T A x \Rightarrow \int xA[x]dx$$ Does any such analogy exist with $ax^4$ type functions? Either in the finite or infinite ...
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### Problem on solving $z^4 - 2z + 4 = 0$ [closed]

I'm having a hard time solving this equation $z^4 - 2z +4 =0$. If you could help me how to solve it that would be amazing and also give me some references of how to solve equations like this ...
3answers
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### Finding the sum of squares of roots of a quartic polynomial.

What is the sum of the squares of the roots of $x^4 - 8x^3 + 16x^2 - 11x + 5$ ? This question is from the 2nd qualifying round of last year's Who Wants to be a Mathematician high school competition ...
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### Factoring quartic into 2 quadratic polynomials: $x^{4}+ax^{3}+bx^{2}+cx+d =(x^{2}+g_{1}x+h_{1})(x^{2}+g_{2}x+h_{2})$

I would like to factor the quartic into two quadratic polynomials $F(g),G(h)$: \begin{align*} x^{4}+ax^{3}+bx^{2}+cx+d & =(x^{2}+g_{1}x+h_{1})(x^{2}+g_{2}x+h_{2}),\\ & =x^{4}+(g_{1}+g_{2})x^{...
2answers
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### How many zeros does the polynomial have in the right half plane?

The polynomial is $f(z) = z^4+\sqrt{2}z^3+2z^2-5z+2$ If you check the image of the imaginary axis, you see that there are no zeros, so we can use the right semicircle from $iR$ to $-iR$,and make $R$ ...
1answer
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### Can we solve $A+D\sin^{2}x=B\sin x+C\cos x$ without having to solve a quartic polynomial?

Suppose the following equation $$A+D\sin^{2}x=B\sin x+C\cos x,$$ where $A,B,C,D\in\mathbb{R}$ are the real constants. Initially, I tried to find its solution from a simple substitution \begin{align*}...
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### Finding the minimum of a multivariate quartic function over a unit sphere.

Finding the minimum of a quadratic function over a unit sphere is simple I believe. You just have to find the minimum eigenvalue of the coefficient matrix. I would like to find the minimum of a ...
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### Arithmetical progression and quadratic equation

Determine the real number $k$ with the condition that the roots of the equation $x^ {4}-(3k+2) x^ {2} +k^ {2} =0$ make the arithmetic progression? I dont know how to start ?
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### Median Age of Women at First Marriage ; Linear Algebra application problem

We are given an application problem: The median age of women in the United States at first marriage is given below. Let t= 0 correspond to 1970 and also let tbe measured in decades. Let 0 ≤ t≤ 4. ...
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### How to solve that?

I have no idea about how to solve the following: $$\sqrt{13x+1} + \sqrt{4x-1} = 3\sqrt{x}$$ Could somebody help me, please?