# 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.

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### Let $x_1,x_2,x_3,x_4$ denote the four roots of the equation $x^4 + kx^2 + 90x - 2009 = 0$. If $x_1x_2 = 49$, then find the value of $k$.

Let $x_1,x_2,x_3,x_4$ denote the four roots of the equation $$x^4 + kx^2 + 90x - 2009 = 0.$$ If $$x_1x_2 = 49,$$ then find the value of $k$. What I tried :- From Vieta's Formula for quartic equations ...
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### need help solving a quartic equation

The question I am asking is to solve the equation $x^4-4x-1=0$, I need an exact answer. What I have done was found out that it equals $(x^2+1)^2 - 2(x+1)^2 =0$. Anybody help me, please?
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### How can I find the roots of the polynomial $12x^{4}+2x^3+10x^2+2x-2$?

It's clear that I can divide by $2$, but I don't know what can I do with $$6x^{4}+x^3+5x^2+x-1$$ Is there any algorithm for it or a trick? I have found the roots by an online calculator but I don't ...
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### Geometric sequence problem including sum of the numbers

Numbers: $a,b,c,d$ generate geometric sequence and $a+b+c+d=-40.$ Find these numbers if $a^2+b^2+c^2+d^2=3280$ I tried this problem and I have system of equations which I can't solve. I think there ...
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### Beyond homography: from conics to quartics

It is known that a homographic transformation of a circle is a conic. The first figure is an example that fix the notation. Here the point $O$ is the center of the homography $h$. The axis of the ...
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### Is there a quick (hopefully elementary) way to prove that $6b^2c^2 + 3c^2 - 36bc - 4b^4 - 4b^2 + 53=0$ has only one solution?

I have the Diophantine equation $$6b^2c^2 + 3c^2 - 36bc - 4b^4 - 4b^2 + 53=0.$$ Numerical calculations suggest this has only one positive integer solution, namely $(b,c)=(2,3)$. Is there a quick way ...
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### Basic Geometry Problem : FInd $AC$

Let $ABC$ be a triangle where $\angle B$ is a right angle. Extend $AC$ up to point $D$ such that $\angle CBD=30^\circ$. If $AB=CD=1$, find $AC$. My approach to this problem is first to try to find ...
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### Find roots of quartic equation $x^4-x+1=0$

How to solve $$x^4-x+1=0$$ My attempt: $$x^4-x+1=0$$ $$\implies x^4-x^3-x+1+x^3=0$$ $$\implies x^3(x-1)-(x-1)+x^3=0$$ $$\implies (x^3-1)(x-1)+x^3=0$$ But I couldn't find a way to combine $x^3$ into ...
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### Can someone help me solve this quartic equation?

$$x^4+5x^3-18x^2-10x+4=0$$ I cannot solve this quartic equation - is there any way to solve it apart from the quartic equation? It has no integer roots, and a hint given on the worksheet says to ...
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### Quartic Formulation of tan(x) = x

I want to convert the above equation into quartic form. The only thing I can think of is to use the Taylor series expansion of tan(x). Is there any other way, anyone can suggest to convert the above ...