For equations in which the variable(s) is/are under a radical.

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### free software for radical algebraic equations

I want to study an algebraic curve defined by equations of the form $$a_1 \sqrt{f_1(x)} + ... + a_n \sqrt{f_n(x)} = 0,$$ where $x$ is a real variable and $f_i$ are polynomials. $a_1,... a_n$ could ...
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### How to interpret an expression when the radical doesn't extend over anything?

I have a school assignment which includes solving this problem from a scanned document: Equivalent: Given that $m = { \sqrt{} l - n^2 \over n }$, express $n$ in terms of $m$. How do interpret this ...
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How would you solve this problem for real $x$? $$\sqrt{2+\sqrt{2-x}}=\sqrt{x-\frac 1x} + \sqrt{1-\frac 1x}$$ It can be easily shown that both equations $$x=\sqrt{2+\sqrt{2-x}}\tag{1}$$ and $$x=\sqrt{... 0 votes 0 answers 25 views ### Show the radical axis of any two circles in the family is the line x+y=0 and show that the circles are real iff -3<λ<0. Show that for λ≠-1 the formula below defines a circle, that the centre lies on the line 3x+y-5=0, and that the radical axis of any two circles in the family is the line x+y=0. Further, show that ... 0 votes 3 answers 61 views ### How to find derivative in radical function? I need to find derivatives of following functions:$$\frac32x^\frac32-\frac{2x^2}{3}-2\sqrt{x}-\frac{-2}{\sqrt{x}}$$So starting from first one, I have tried to first simplify the fractions to ... 0 votes 1 answer 70 views ### How to solve this radical equation for x? Question:$$\frac{x}{\sqrt{x^2+1}} = x^4 - x$$I tried:$$\rightarrow \frac{1}{\sqrt{x^2+1}} = x^ 3 - 1\to\frac{\sqrt{x^2 + 1}+1}{\sqrt{x^2+1}} = x^3$$Now rationalising it$$\to \frac{x^2 +1-1}{... 99 views

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### Pairs of integers $(x,m)$ for which $\sqrt{\sqrt{x-2}+m}+\sqrt{-\sqrt{x-2}+m}=2$ hold?

Find all pairs of integers $(x,m)$ for which $$\sqrt{\sqrt{x-2}+m}+\sqrt{-\sqrt{x-2}+m}=2$$ hold. I have used this property : Property: if $$a+b+c=0 \implies a^3+b^3+c^3=3abc,$$ I come ...
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### Solvable elements of a field extension

Suppose $K$ over $F$ is a field extension, and $\alpha \in K$. My instructor says that "$\alpha$ is solvable over $F$ if there exists a radical extension $L$ of $F$ containing $\alpha$". My ...
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### Why substitution in irrational equation doesn't give equivalent equation?

I have two examples of irrational equations: The first example: $\sqrt{3-x} + \sqrt{6+x}=3$ In solution, they take cube of both sides and do following: \begin{eqnarray*} &\sqrt{3-x} &...
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### Solving $\sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$

I have to solve this irrational equation on $\mathbb{R}$ : $$\sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$$ I tried to do a substitution with $u=1-x$ but the only things I manage to reach is the following ...
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### Fast way to solve $4 = \sqrt {x+10}-\sqrt {x-10}$

The question is this: $4 = \sqrt {x+10}-\sqrt {x-10}$ For some reason, I keep on getting 289/3, even though it is the wrong answer. This is from a timed test, and my way is wrong and extremely ...
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### Find the roots for y

$$-1=(0.55)\cdot[1+(y+1)^2]^{\frac{3}{2}}$$ I got stuck with this expression. I have l some difficulty in leanding with some algebraic manipulation. What should I do to solve this equation?? I tried ...
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### simplify the equation $\frac{36}{\sqrt{x}} + \frac{9}{\sqrt{y}} = 42-9\sqrt{x}-\sqrt{y}$

This is the question: $$\frac{36}{\sqrt{x}} + \frac{9}{\sqrt{y}} = 42-9\sqrt{x}-\sqrt{y}$$ This is from a timed competition, and I would like to know the fastest way to do it. I'm not sure, but is ...
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### Semicircle Question

I need help with the question in the image. I just need someone to help by pointing me in the right direction. I don't want a full solution. I want to try to work out this question myself but I just ...
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### Square root with rational exponent

It might seem very stupid question. If $x^2=9$ then to solve for $x$ we take both principal $n$-th root of $9$, i.e. $3$ and the negative $n$-th root of $9$, i.e. $-3$. This is right until I found ...
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### How square roots work in equations?

When I was younger I wasn't paying too much attention or the teacher did not make sure we understood how the square root works. Recently I was faced with some problems where having the right knowledge ...
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### $K=\mathbb{Q}(\sqrt{2},i)$ and let $F=\mathbb{Q}(\sqrt{-2})$. Find the Galois Group $G(K,F)$

Let $K=\mathbb{Q}(\sqrt{2},i)$ and let $F=\mathbb{Q}(\sqrt{-2})$. Identify the $G(K,F)$ with a subgroup of permutations of the roots of $x^8-2$. You have a guideline for the answer in here. But I ...
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### Solve the equation $\sqrt{3\sqrt{x}+1}=\sqrt{2\sqrt{x+1}-1}$

Solve the equation $\sqrt{3\sqrt{x}+1}=\sqrt{2\sqrt{x+1}-1}$. My attempt: With $u=\sqrt{x}, v=\sqrt{x+1}$ I have $u^3=v^2-1$ and $(3u+1)^3=(2v-1)^2$ And I finally have a quadratic equation ...
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### Solve the equation $\sqrt{15-x^3+3x^2-3x}=2\sqrt{x^2-4x+2}+3-x$.

Solve the equation $\sqrt{15-x^3+3x^2-3x}=2\sqrt{x^2-4x+2}+3-x$. I have tried to solve for x by Casio and try to make the equation to $u.v=0$ but the solution is not in $\mathbb{Q}$. Any help is ...
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### How do you know that a positive algebraic radical refers to a nonnegative root?

The online course I am taking says that the 4th root of an equation refers to the nonnegative root (see attached screenshot). But how can you know that it is not a negative root, I thought that that ...
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### Solving a six-degree polynomial of the form $ax^6+bx^3+g$.

I read that it is not always possible to solve but from Wikipedia: Some sixth degree equations, such as $ax^6 + dx^3 + g = 0$, can be solved by factorizing into radicals, but other sextics ... 1 vote
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### Can this equation with multiple radicals be solved using closed form expressions?

For an expression of the following form: $\frac{f(x) + \sqrt{ g(x)}}{h(x)} = \frac{k(x) + \sqrt{ l(x)}}{m(x)}$, where $f(x)$, $g(x)$, $h(x)$, $k(x)$, $l(x)$ and $m(x)$ are all quadratics and where ...
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### Rationalizing radical expressions using conjugates - How does this step work?

This is the full solution given in my book: Can someone please explain to me how it goes from Step 4 to Step 5? Specifically, I do not understand how the numerator simplifies to -1 and how the first ...
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I've been trying to solve this one without success... can anybody help me? The result should be $x=\frac{17}{16}$ and it's correct, I've already checked. This is the equation: $$\frac{1}{\sqrt {x+2}... 0 votes 2 answers 75 views ### solving radical equation$$\sqrt{3x+1} - \sqrt{6-x} +3x^2-14x-8=0 $$I tried : 3x+1 = a  , 6-x=b  and tried to make  3x^2-14x-8  to be in term of a and b, I'm unable to solve it so far. but I know the answer is ... 0 votes 1 answer 182 views ### How to solve this equation with rational exponents? I've been really struggling to solve this one, could you provide how you'd solve it?$$3x^{2/3} + 4x^{1/3} =4
I am trying to understand construction of irreducible polynomial of odd degree over $\mathbb{Q}$ with exactly two non-real roots. Let $g(x)=(x^2+m)(x-n_1)\cdots (x-n_{k-2})$ with $m>0$, \$n_1< \...