Questions tagged [radical-equations]

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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2 votes
1 answer
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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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1 vote
3 answers
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A golden question $\sqrt{2+\sqrt{2-x}}=\sqrt{x-\frac 1x} + \sqrt{1-\frac 1x}$

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{...
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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 ...
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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 ...
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1 answer
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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}{...
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2 votes
1 answer
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Understanding Cardano's Formula

In deriving his formula, Cardano arrives at the equation $y^3+py+q=0$. By substituting $y=\sqrt[3]{u}+\sqrt[3]{v}$, he gets the equation $(u+v+q)+(\sqrt[3]{u}\sqrt[3]{v})(3\sqrt[3]{u} \sqrt[3]{v} +p)=...
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3 votes
5 answers
312 views

Solving $(x+1)\sqrt{2(x^2 + 1)} + \sqrt{6x^2 + 18x +12} = \frac{3x^2 + 7x + 10}{2}$

Today, I came across this problem. $$(x+1)\sqrt{2(x^2 + 1)} + \sqrt{6x^2 + 18x +12} = \dfrac{3x^2 + 7x + 10}{2}$$ We are asked to find the possible values of $x$ satisfying this equation. The first ...
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Expressing the solution to a rational expression in radical form

I'm trying to find the solution to this equation $$ -\frac{3}{r} + \frac{8}{r^3} = \frac{\sqrt{2}-1}{2} $$ but I haven't been able to find a solution in radical form. Although I've found the solutions ...
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Why can't a negative base be raised to a non-integer exponent?

Why can't we do this? Example: $(-1) ^ {1/3}$ Math definitions are based on a definite logic. What is the logic here? Can you give me some examples where it violates the equation? I'm just a high ...
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5 votes
3 answers
206 views

If $\sqrt{a}\sqrt{b}=\sqrt{ab}$ only holds for positive real $a$ & $b$, then why can we say $\sqrt{-a}=\sqrt{-1\cdot a}=\sqrt{-1}\sqrt{a}=i\sqrt{a}$?

I am a little bit bummed that I have this question as I'm sure it has been asked before (I couldn't find the answer) but... If $\sqrt{a}\sqrt{b} = \sqrt{ab}$ is only true for positive reals $a$ and $...
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6 votes
4 answers
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Solve for $x$ in $\sqrt{x+2\sqrt{x+2\sqrt{x+2\sqrt{3x}}}} = x$

Solve for $x$: $$\sqrt{x+2\sqrt{x+2\sqrt{x+2\sqrt{3x}}}} = x$$ I tried to substitute $y=x+2$ and then I try to solve the equation by again and again squaring. Then I got equation, $$(y-2)(3y^{14}-(y-...
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5 votes
4 answers
228 views

Does this root technically count as a solution to this radical equation?

$$x=\sqrt{2x+3}$$ If you solved this traditionally you would get $x_1=3$ & $x_2=-1$. But inputting $x=-1$ in $\sqrt{2x+3}$ gives $+1$ or $-1$. The original equation is only valid if $\sqrt{2x+3}=-...
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3 votes
4 answers
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Solving $\sqrt[3]{x-3}+\sqrt[3]{1-x}=1$

The Equation How can I analytically show that there are no real solutions for $\sqrt[3]{x-3}+\sqrt[3]{1-x}=1$? My attempt With $u = -x+2$ $\sqrt[3]{u-1}-\sqrt[3]{u+1}=1$ Raising to the power of $3$ $$(...
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3 votes
2 answers
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Pairs of integers $ (x,m)$ for which $\sqrt[3]{\sqrt[3]{x-2}+m}+\sqrt[3]{-\sqrt[3]{x-2}+m}=2$ hold?

Find all pairs of integers $(x,m)$ for which $$\sqrt[3]{\sqrt[3]{x-2}+m}+\sqrt[3]{-\sqrt[3]{x-2}+m}=2$$ 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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1 vote
0 answers
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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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Calculate the sum of all irrational roots of $4\sqrt[3]{8x- 3}- 8x^{3}- 3= 0$

Calculate the sum of all irrational roots of $$4\sqrt[3]{8x- 3}- 8x^{3}- 3= 0$$ I'm not even sure how to begin here, I tried raising it to the power of three, tried writing $8x^{3}+ 3$ with $x^{3}+ y^...
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1 vote
3 answers
408 views

The set of all $x$ satisfying, $\sqrt{4x+1} + \sqrt{7-x} = 6 $, consists of:

The set of all $x$ satisfying, $\sqrt{4x+1} + \sqrt{7-x} = 6 $, consists of: $A)$ Two rational numbers. $B)$ An irrational number. $C)$ Complex number. $D)$ None. How to solve the above question ...
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3 answers
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How many zeros does a radical equation (eg, $X^{4/3}-5X^{2/3}+6=0$) have?

I want to know if there is a general rule that will give me the answer. I'm not talking about crazy expressions under a radical, just simple variables raised to a fractional exponent like: $$X^{4/3}-...
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2 votes
8 answers
318 views

Solve the equation $x+\frac{x}{\sqrt{x^2-1}}=\frac{35}{12}$ [closed]

Solve the equation $$x+\dfrac{x}{\sqrt{x^2-1}}=\dfrac{35}{12}.$$ The equation is defined for $x\in\left(-\infty;-1\right)\cup\left(1;+\infty\right).$ Now I am thinking how to get rid of the radical in ...
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1 vote
4 answers
113 views

Solve the equation $\sqrt{45x^2-30x+1}=7+6x-9x^2$

Solve the equation $$\sqrt{45x^2-30x+1}=7+6x-9x^2.$$ So we have $\sqrt{45x^2-30x+1}=7+6x-9x^2\iff \begin{cases}7+6x-9x^2\ge0\\45x^2-30x+1=(7+6x-9x^2)^2\end{cases}.$ The inequality gives $x\in\left[\...
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2 votes
4 answers
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Solve the equation $\sqrt{x^2-1}=(x+5)\sqrt{\frac{x+1}{x-1}}$

Solve the equation $$\sqrt{x^2-1}=(x+5)\sqrt{\dfrac{x+1}{x-1}}.$$ I think that radical equations can be solved by determining the domain (range) of the variable and at the end the substitution won't ...
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3 answers
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Solve $\frac{7}{x+\sqrt{x+5}}+\frac{7}{x-\sqrt{x+5}}=8$

Solve the equation: $$\dfrac{7}{x+\sqrt{x+5}}+\dfrac{7}{x-\sqrt{x+5}}=8.$$ I am not sure how to approach the problem. Should we first determine the domain? I think we can also check for every value we ...
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0 votes
0 answers
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Solving logarithmic inequality and omitting the condition for positive numerus

Irrational equalities of the form $\sqrt{f(x)} \geq g(x)$ are equivalent to $( f(x) \geq 0, g(x) \geq 0, f(x) \geq (g(x))^2)$ or $( f(x) \geq 0 $ and $g(x)<0)$ . In some sources I found that ...
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1 answer
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How To Solve $\frac{1}{X}\bigg\lfloor-\frac{3}{2}+\frac{1}{2}\sqrt{8X+9}\bigg\rfloor=\frac{1}{N}$ over the integers.

Question: If $X$ and $N$ are positive numbers. How would I solve for $X$ in the following equation: $$ \frac{1}{X}\bigg\lfloor-\frac{3}{2}+\frac{1}{2}\sqrt{8X+9}\bigg\rfloor=\frac{1}{N} \label{a}\tag{...
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1 vote
1 answer
51 views

Can fractional/decimal radicals/roots exist?

For questions like "What is the 1/2th root of x would the answer be $x^2$? My logic is that since $$ \sqrt[\cfrac{1}{2}]{x}=x^{1/{(\cfrac{1}{2}})} $$ Which simplifies to $x^2$. So as a general ...
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2 answers
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Equivalence of two radical equations without certain conditions - correctness of method

I have question related to the following example: $\sqrt{22-x} - \sqrt{10-x}=2$. First question: Do I first need conditions $22-x \geq 0 $ and $ 10-x \geq 0$ to obtain the equivalent equation: $\sqrt{...
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2 votes
2 answers
96 views

Why substitution in irrational equation doesn't give equivalent equation?

I have two examples of irrational equations: The first example: $\sqrt[3]{3-x} + \sqrt[3]{6+x}=3$ In solution, they take cube of both sides and do following: \begin{eqnarray*} &\sqrt[3]{3-x} &...
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6 votes
1 answer
534 views

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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4 votes
3 answers
130 views

Fast way to solve $4 = \sqrt[3] {x+10}-\sqrt[3] {x-10}$

The question is this: $4 = \sqrt[3] {x+10}-\sqrt[3] {x-10}$ 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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  • 533
0 votes
1 answer
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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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1 vote
1 answer
74 views

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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5 votes
5 answers
221 views

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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3 votes
2 answers
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Solving $\sqrt[3]{x+1} - \sqrt[3]{x-1} = \sqrt[3]{x^2-1}$ for real $x$

Solve the equation in the Real number system: $$\sqrt[3]{x+1} - \sqrt[3]{x-1} = \sqrt[3]{x^2-1}$$ I have attempted using $(A-B)^3 = A^3 - B^3 - 3.A.B.(A-B)$ with $A = \sqrt[3]{x+1}$ , $B = \sqrt[3]{x-...
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-1 votes
1 answer
101 views

Integrate $\frac{x}{(x^4+1)^2 \sqrt{(x^2-x+1)}}$. [closed]

Evaluate the indefinite integral, $$\int\frac{x}{(x^4+1)^2 \sqrt{(x^2-x+1)}} \mathrm{d}x$$ Found this problem in a mathematics group site, but the solution was never posted. I suspect it cannot be ...
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2 votes
1 answer
79 views

Zero set of nested radicals

My question deals with a function on $\mathbb{R}^n$ that consists of nested radicals and polynomial functions. I'm not even sure how to properly formulate this question, i.e. precisely what class of ...
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  • 158
1 vote
1 answer
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Determining the extraneous solution to a radical equation

Let's say I am trying to solve the equation $ax -b\sqrt{x}=c$ such that $a,b,c>0$. Rearranging, squaring and using the quadratic equation yields the solutions $x^*=\frac{2ac+b^2 \pm b\sqrt{b^2+4ac}}...
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  • 143
0 votes
1 answer
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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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0 votes
2 answers
218 views

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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  • 103
2 votes
1 answer
164 views

$K=\mathbb{Q}(\sqrt[8]{2},i)$ and let $F=\mathbb{Q}(\sqrt{-2})$. Find the Galois Group $G(K,F)$

Let $K=\mathbb{Q}(\sqrt[8]{2},i)$ 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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0 votes
1 answer
63 views

Solve the equation $\sqrt{3\sqrt[3]{x}+1}=\sqrt[3]{2\sqrt{x+1}-1}$

Solve the equation $\sqrt{3\sqrt[3]{x}+1}=\sqrt[3]{2\sqrt{x+1}-1}$. My attempt: With $u=\sqrt[3]{x}, v=\sqrt{x+1}$ 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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5 votes
1 answer
184 views

Solve the equation $\sqrt[3]{15-x^3+3x^2-3x}=2\sqrt{x^2-4x+2}+3-x$.

Solve the equation $\sqrt[3]{15-x^3+3x^2-3x}=2\sqrt{x^2-4x+2}+3-x$. 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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0 votes
2 answers
54 views

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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1 vote
1 answer
83 views

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 ...
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1 vote
2 answers
120 views

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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1 vote
2 answers
21 views

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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1 vote
1 answer
45 views

Irrational equation high school

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}...
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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 ...
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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? $$3x^{2/3} + 4x^{1/3} =4$$
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  • 43
0 votes
1 answer
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Constructing (irreducible) polynomial of odd degree with exactly two non-real roots

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< \...
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