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.

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What's wrong with manipulating this algebraic equation? and why does a manipulated system of equations have a different solution than the original?

I'll give an example for my first question: $x^2 + x + 1 = 0$ Clearly $x = 0$ and $x = 1$ aren't solutions, so first we can safely divide by $x$: $x + 1 + 1/x = 0$ By subtracting $1/x$ from both sides we get: $x + 1 = -1/x$ By plugging the value $x…
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Solve $2x^2+y^2-z=2\sqrt{4x+8y-z}-19$

I am trying to solve the following equation. $$ 2x^2+y^2-z=2\sqrt{4x+8y-z}-19 $$ To get rid of the square root, I tried squaring both sides which lead to $$ (2x^2+y^2-z+19)^2=16x+32y-4z $$ which was too complex to deal with. Also, I have tried some…
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Roots of a certain sixth order polynomial

I am looking for the roots (or basically any information regarding them) of the sixth order polynomial $$p(x):=ax^6+(a+1)x^4+2bx^3-b^2$$ for positive, real constants $a,b$. Since $p(0)=-b^2<0$ and $\lim_{x\to\pm\infty}p(x)=+\infty$ we have at least…
weee
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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 then I'd simplify, but it's still very hard to…
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Applying the quadratic Tschirnhausen transformation

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…
Simply Beautiful Art
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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 the form $$y=x^2+\alpha x+\beta$$ which, by a…
user65203
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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…
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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 that I am a student in physics, not in mathematics,…
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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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If $\sum_{i=k}^n {n \choose i} p^{i}(1-p)^{n-i} \approx 0.05$, how can we find $k$?

Let $n$ be any natural number, let $k\in\{0, \dots, n\}$, and let $p \in [0, 1]$. If $\sum_{i=k}^n {n \choose i} p^{i}(1-p)^{n-i} \approx 0.05$, how can we find $k$ (in terms of $n$ and $p$)?
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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 eigenvalue. ( $M$ is invertible, and…
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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 be other solutions). Does anybody have a good (or…
user474282
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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…
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Proving $2((a+b)^4+(a+c)^4+(b+c)^4)+4(a^4+b^4+c^4+(a+b+c)^4)=3(a^2+b^2+c^2+(a+b+c)^2)^2$ in another way?

How do I prove the following identity without expanding both sides directly. $$2((a+b)^4+(a+c)^4+(b+c)^4)+4(a^4+b^4+c^4+(a+b+c)^4)\\=3(a^2+b^2+c^2+(a+b+c)^2)^2$$ I expanded both sides directly and it is true. However, I was hoping there could be…
Shuaib Lateef
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How to solve Quadratic Equations with an Unknown C and other variables.

For instance $3x^2 - 11x + r$, I understand the value of $r$ is $6$ through trial and error but trial and error is extremely inefficient and time consuming thus not useful in exam situations, How would I solve this? And also : If $x^2 +px + q$ is a…
Davidovich
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