Mathematics

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.

77 questions
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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…
Abdullah Alhussni
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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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Is the number system for $x$ assumed beforehand when proving the quadratic formula?

When proving the quadratic formula (or any other mathematical equation, definition, formula, etc., from like all the way from basic math to advanced calculus), do we have to assume/declare the number system of $x$ beforehand, or do we determine that…
Aaditya Visavadiya
  • 359
5
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2 answers

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…
watermelon7288
  • 61
5
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1 answer

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 solve.…
Heart
  • 735
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1 answer

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
4
votes
4 answers

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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1 answer

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 following:…
Simply Beautiful Art
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3
votes
2 answers

Do we have to assume $x$ exists before starting to solve algebraic equations?

When solving algebraic equations (for example, $2x+6=10$ , $x+5=x$ , or $x^2=-1$ ), do we have to assume that our unknown variable ( $x$ ) exists (i.e., it exists in a specific number system) before solving? Like before we solve or rearrange…
Aaditya Visavadiya
  • 359
3
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2 answers

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?
CN_lift
  • 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
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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
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3
votes
1 answer

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,…
Barbaud Julien
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2
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1 answer

Extraneous solutions in algebraic equations

Consider these 2 Equations , where multiplication by $x$ is involved. It is known that multiplication by $x$ should introduce the extraneous solution $x=0$ in both cases. Observe that the second equation the unwanted $x=0$ solution appears, while…
mawaior
  • 385
2
votes
1 answer

Rewriting $||x-y|-z|$, for non-negative $x$, $y$, $z$, in a way that does not involve nested absolutes

Is there an idea that resolves nested value expressions into several separate expressions each using a single absolute value? Something similar like resolving the max function using the absolute value. More concretely, I am looking for a way to…
chickenNinja123
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