Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [algebraic-equations]

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

An algebraic equation is an equation of the form $P = 0$ where $P$ is a polynomial with coefficients in some field, often the field of rational numbers.

For most authors, an algebraic equation is univariate, which means that it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called multivariate and polynomial equation is usually preferred to algebraic equation. For example, $$x^5 - 3x + 1 = 0$$ is an algebraic equation with integer coefficients, and $$y^4 +\frac{xy}2 = \frac{x^3}3 - xy^2 + y^2 - \frac17$$ is a multivariate polynomial equation with rational coefficients.

67 questions
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Prove that x = y if $(\sqrt{y^2+x}+x)(\sqrt{x^2+y}-y)=y$

The problem is: For real numbers x and y, if: $(\sqrt{y^2+x}+x)(\sqrt{x^2+y}-y)=y$ then $x=y$. Firstly, I prove that for $y=0$, we have $x=0$. Then I make $x=ky$ for real $k$. =>$(\sqrt{y^2+ky}+ky)(\sqrt{k^2y^2+y}-y)=y$ <=>…
Kii
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How to solve this algebraic equation?

$$x^{\cfrac{1}{x}}=(1+x)^{\cfrac{1}{1+x}}$$ Domain: $(0,+\infty)$ I know a numerical solution of $x\approx 2.293$ Does it have any analytical solutions? If not, is it possible to prove that it doesn't have any analytical solutions?
SundayLi
  • 184
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Combine ODE with constraints using a Lagrange multiplier

Consider a constrained ODE system: \begin{align} \dot{\bf x} &= \bf f(t,\bf x), \\ st. 0 &= \bf g(t,\bf x). \end{align} I wish to combine these into a single equation using a Lagrange multiplier, however I am unsure how to do it. I talked with my…
Tue
  • 207
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Is there a better way to solve this equation?

I came across this equation: $x + \dfrac{3x}{\sqrt{x^2 - 9}} = \dfrac{35}{4}$ Wolfram Alpha found 2 roots: $x=5$ and $x=\dfrac{15}{4}$, which "coincidentally" add up to $\dfrac{35}{4}$. So I'm thinking there should be a better way to solve it than…
Kurumi Gaming
  • 187
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Need help simplifying a set of equations (and understanding how to solve it)

i have three algebraic expressions, each using the others. in these equations a, b, c and t are known and plugged in later: $x = a^{-1}(t + y + z)$ $y = b^{-1}(t + x + z)$ $z = c^{-1}(t + x + y)$ i have managed to successfully solve the equations…
AceldamA
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Find the value of x in the equation below

$\dfrac{3+x}{2022}+\dfrac{2+x}{2023}+\dfrac{1+x}{2024}+\dfrac{x}{2025}=-4$ $(S:x=-2025)$ Solving in the usual way we find the solution. I would like to know if there is a more practical way of solving this without using least common multiple.
peta arantes
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Solving a system of quadratic equations in nine variables

I'm trying to carry out the forward displacement analysis of a parallel mechanism. This question is simplified as follows. Let $Q_1$, $Q_2$, $Q_3$, $A_1$, $A_2$, $A_3$ be six unit vectors that pass the origin. $Q_1$, $Q_2$, $Q_3$ are variables, and…
Jinping Zhang
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What is the exact solution for this equation?

I have been thinking about this equation: $$x^2=2^x$$ I know there is two integer solutions: $x=2$ and $x=4$. But there also is a negative solution, that is approximately $x=-0.77$. $$(-0.77)^2=0.5929$$ $$2^{(-0.77)}=0.5864...$$ Can we find this…
Roberto Gandulfo
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Show that, if a, b, c are real numbers and ac = 2(b + d), then at least one of $x^2 + ax + b = 0$ and $x^2 + cx + d = 0$ has real roots.

Show that, if $a, b, c$ are real numbers and $ac = 2(b + d)$, then, at least one of the equations $x^2 + ax + b = 0$ and $x^2 + cx + d = 0$ has real roots. I've have tried many times and used different methods but can't prove it.
user762306
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Does this equation imply separation of variables?

I have the following equation: \begin{equation} \dfrac{q(t,r)}{s(t,r)}=B(r), \qquad \qquad (1) \end{equation} with $q>0$ and $s>0$. Apart from the trivial case $q(t,r)=\mathrm{const.} \times s(t,r)$ (i.e., $B(r)= \mathrm{const.}$) does this equation…
Frank
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Find positive integer solutions of cubic equation with three variables

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 is no simplification on this way. Then I've tried…
Michael Galuza
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Show that $\frac {1}{\sqrt{5}}[(\frac {1}{x+r_+}) - (\frac {1}{x+r_-}) = \frac {1}{\sqrt{5}x}[(\frac {1}{1-r_{+}x}) - (\frac {1}{1-r_{-}x})] $

I need to manipulate this equation: $$ \frac {1}{\sqrt{5}}\left(\frac {1}{x+r_+} - \frac {1}{x+r_-}\right) $$ to show that $$ \frac {1}{\sqrt{5}}\left(\frac {1}{x+r_+} - \frac {1}{x+r_-}\right) = \frac {1}{\sqrt{5}x}\left(\frac {1}{1-r_{+}x} -\frac…
Leyla Alkan
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0
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0 answers

Algebraic equations and geometrical shape construction by a straightedge and compass

In his book, The Annotated Turing, Charles Petzold says (emphasis mine), Using a straightedge and compass to construct geometrical shapes is equivalent to solving certain forms of algebraic equations. Does that mean that given any polynomial…
Kedar Mhaswade
  • 289
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1 answer

Resolve 1 equation with 3 unknowns with specific condition for each unknown

I want to find the possible solution of the following problem: $T_k= 21600$ seconds $1281.49 < T < 21600 $(T in seconds) $x$ and $y$ don't have unit $x$ and $y$ have to be whole number $x > 0$ and $y > 0$ $$0.5(2x+1)*T-0.5(2y+1)*T_k=0$$ I don't know…
Alexandre Fargeix
  • 11
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vertical displacement

I have the below HW question: What I've done is replace t with 2, thus obtaining that the ball is at a height of 1.25m after 2 seconds and hence, it will travel over the net. Am I correct?
Joann Sammut
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