Questions tagged [algebraic-equations]

Use this tag for questions related to solving equations involving polynomials.

25 questions
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Algebraic equation of first degree with absolute values [closed]

How to solve $$\left| a^2-2 a-b^2-4 b-x \right| + \left| a^2-2 a-b^2-4 b-3 x+2\right| +\\ \left| a^2-2 a+b^2+4 b+2 x \right| + a^2-2 a+b^2+4 b+18 \left| x-2 \right| +11 x=20$$ over the reals if ...
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Solve the equation for x.

solve the following equation for x $(x^2-8x-3)÷(8x-3)=(x^2+4x+4)÷(4x+4)$ The problem here I am encountering with is figuring out which method to be used here for solving. I am not getting how to ...
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Solve $(1+x)^2=A\sqrt{1+Cx}$ for $x$.

$x>0$, $A>0$ and $C>1$. I am trying to come up with a closed form expression for $x$, even if it is an approximation. Any help appreciated.
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how to handle a “Stiff” algebraic equation numerically?

I have a question of great practical importance for me, but I would like to ask it on a bit more of a theoretical mode, because I feel I lack the basic knowledge on it. I would also like to mention ...
119 views

Solving $8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$

Solve the equation $$8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$$ I have this idea: set $$\sqrt{x+2}=a , x+2=a^2 , \sqrt{x-1}=b.$$ So $$x-1=b^2 , 2a^2+6b^2 =8b-4$$ and $$x^2+x-2 =a^2b^2$$ and ...
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If $x$ is algebraic over a quotient field $K$ of $A$, then there exists an integral element $cx$ for some $A \ni c \neq 0$.

Let $A$ be a commutative ring, $K$ its quotient field and $x$ algebraic over $K$. This means that there exists a polynomial $f(X)$ with coefficients in $K$ such that $f(x) = 0$. In other words, if ...
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upper and lower limits for finding just one (real) solution of an algebraic equation (degree 5 and lower)

While programming an equation solver (using trial and error), I came across the fact that there are multiple real and complex bounds in which all of the solutions should be. For my case, only real ...
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Solution to $Mx-x^2=0$ where $x^2$ is the square of the elements of vector $x$

I have been trying to find the solutions for $$Mx=x\circ x$$ where $\circ$ is the element wise product. One solution is $x=0$. But there is another solution $x\neq 0$, if $M$ has a real positive ...
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Solving algebraic equations involving factorials - without trial and error.

The question is defined as follows: $$\frac{(x!)^3}{x}-1=3455$$ I first did the basics which was getting rid of the 1 onto the left and getting rid of the $x$, then I factorised both sides to get an ...
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As per my previous question, I attempted to take dxiv's approach, though I can't seem to make much headway. Considering the simpler problem $x^3=x+a$ and the substitution $y=x^2+mx+n$, I got the ...
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Proving non-existence of real roots

Prove that for any real numbers $a_{85}, a_{84}, a_{83},\dots ,a_3$ the equation $a_{85}x^{85}+ a_{84}x^{84}+ a_{83}x^{83}+\dots +a_3x^3+3x^2+2x+1=0$ has no real roots. (This problem is stated in ...
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Solving a symmetric equation involving three variables a,b and c

Solve : $$\frac{x+a^2}{a+b}+\frac{x+b^2}{b+c}+\frac{x+c^2}{c+a}=2(a+b+c)$$ I am trying to find a simple technique to solve this equation as there is a pattern in the equation, but I could not do. Any ...
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approximating irrational roots of algebraic equations with the Pierce expansion

Let $p (x) = 1 - x \lfloor \frac{1}{x} \rfloor$ then the Pierce expansion of a real number $x \in {R}$ is expressed by \begin{equation} x_1 = \sum_{n = 1}^{\infty} (- 1)^{n + 1} \prod_{m = 1}^n ...
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Solving $x^2 y''(x) + 6x y'(x) - 9y(x) =0$ with similar techniques that are used to solve algebraic equations

Consider the ODE $$x^2 y''(x) + 6x y'(x) - 9y(x) =0 .$$ It is clear that we can solve the ODE by the method of reduction of order. However, if we "see" the function $y$ as some constant just for a ...
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Explanation of the Tschirnhausen transformation

I am studying the resolution of the quintic equations, which involves the so-called Tschirnhausen transform. The idea is to cancel the fourth and third degree coefficients by a change of variable of ...
196 views

How to find all nonnegative integers $x, y, z$ and $w$ such that $2^x3^y-5^z7^w=1$

Find all nonnegative integers $x, y, z$ and $w$ such that $2^x3^y-5^z7^w=1.$ I think they are $(x,y,z,w)=(1,0,0,0),(1,1,1,0),(3,0,0,1),(2,2,1,1)$, but I couldn't prove its sufficiency (or there may ...
Find positive integer solutions of equation $$t^3 -at^2 + bt - c=0,$$ where $$a^3-6ab+7c=0$$ ($a, b, c$ are positive integers too). I've tried to use find solutions in modular arithmetic, but there ...