Questions tagged [radical-equations]

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

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3answers
50 views

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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8answers
135 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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4answers
66 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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4answers
93 views

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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3answers
84 views

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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0answers
70 views

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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1answer
58 views

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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1answer
35 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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2answers
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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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2answers
74 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} &...
6
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1answer
113 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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3answers
120 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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1answer
39 views

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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1answer
39 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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5answers
149 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 ...
3
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2answers
106 views

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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1answer
95 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 ...
2
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1answer
67 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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1answer
54 views

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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1answer
35 views

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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2answers
37 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 ...
2
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1answer
52 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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1answer
52 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 ...
4
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1answer
121 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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2answers
43 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 ...
1
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1answer
64 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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2answers
113 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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2answers
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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1answer
42 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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2answers
66 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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1answer
77 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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1answer
99 views

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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0answers
22 views

Radical Equation with Symmetry

Is there an approach to solve the following radical equation in $x$? $(P - 1)x = QR\left( Q + \sigma \sqrt{Q^2 + x} \right)\left( R + sign(P-1) \sigma \sqrt{R^2 + x} \right)$ We know $x\in (0,1)...
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2answers
64 views

Confusing regarding a nested radical equation

Consider the following For all $a\in\Bbb R$ solve the equation $$\sqrt{x^2+4a^2\sqrt{x+a}}=x+2a$$ It is immediate to see that we got the restriction $x\geqslant-a$ (even though not given I assume ...
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1answer
70 views

If $f(x) \Bbb Q[x]$ has splitting field of degree $16$ over $\Bbb Q$ , are the roots of $f(x)$ solvable by radicals.

If $f(x) \Bbb Q[x]$ has splitting field of degree $16$ over $\Bbb Q$ , are the roots of $f(x)$ solvable by radicals. My attemt: Any general equation of degree $\geq 5$ is not solvable by radicals. ...
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5answers
259 views

Solve for $x$ : $\sqrt{x-6} \, + \, \sqrt{x-1} \, + \, \sqrt{x+6} = 9$?

I want to solve the following equation for $x$ : $$\sqrt{x-6} \, + \, \sqrt{x-1} \, + \, \sqrt{x+6} = 9$$ My approach: Let the given eq.: $$\sqrt{x-6} \, + \, \sqrt{x-1} \, + \, \sqrt{x+6} = 9 \...
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1answer
58 views

Rationalize irrational equation

$$ \begin{cases} \sigma_{1,(2)}=x_1+x_2\\ \sigma_{2,(2)}=x_1x_2\\ \end{cases}\\ \color{red}{ \begin{align*} &&p_2&=\sqrt{x_1}+\sqrt{x_2}\\ &\Rightarrow&\left({p_2}^2-\sigma_{...
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2answers
52 views

How to solve a limit through rationalizing but the radicand is a linear relation and the denominator is a quadratic? [closed]

The question is : $$\lim_{x\to 0}{\frac{\sqrt {3x+4}-6}{x^2}}$$ My friend and I are absolutely stumped and can only attempt to solve through substitution. Any ideas?
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3answers
50 views

Solving a radical function [closed]

I already did the previous algebra and i'm on this one step that I can't seem to get past. $$6^{\frac16}\cdot\frac{2x^8}{15}=2x^b$$ solve for b thank-you
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4answers
94 views

If $a+\sqrt{a^2+1}= b+\sqrt{b^2+1}$, then $a=b$ or not?

It might be a silly question but if $$a+\sqrt{a^2+1}= b+\sqrt{b^2+1},$$ then can I conclude that $a=b$? I thought about squaring both sides but I think it is wrong! Because radicals will not be ...
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3answers
32 views

Equations involving squaring a variable under a radical sign

If anyone can help me with how to go about solving these kind of equations i would really appreciate it. :-) $$\sqrt{36-2x^2} = 4$$ Solve for X
4
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3answers
62 views

Irrational equation

Solve over the real numbers: $$(x^2+x+1)^{1/3}+(2x+1)^{1/2}=2$$ I know for the second radical to be defined $x≥-0,5$ and I've attempted various methods I've solved other such equations with but to no ...
5
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3answers
221 views

Solve $\sqrt{\frac{4-x}x}+\sqrt{\frac{x-4}{x+1}}=2-\sqrt{x^2-12}$

The equation is $$\sqrt{\frac{4-x}x}+\sqrt{\frac{x-4}{x+1}}=2-\sqrt{x^2-12}$$ I tried squaring both left side and right side then bringing them to same numerator but got lost from there ... any ...
5
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2answers
148 views

Finding value of $\lim\limits_{n\rightarrow \infty}\Big(\frac{(kn)!}{n^{kn}}\Big)^{\frac{1}{n}}$

Finding value of $\displaystyle \lim_{n\rightarrow \infty}\bigg(\frac{(kn)!}{n^{kn}}\bigg)^{\frac{1}{n}}$ for all $k>1$ Try: I have solved it using stirling Approximation $\displaystyle n!\approx ...
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3answers
96 views

Why can't we remove the radicals from $ \sqrt a + \sqrt b = 8 $ by writing $(\sqrt a)^2 + (\sqrt b)^2 = 8^2$?

Given the equation: $$ \sqrt a + \sqrt b = 8 $$ Why is it wrong to remove the radicals like this? $$\sqrt a + \sqrt b = 8 $$ $$(\sqrt a)^2 + (\sqrt b)^2 = 8^2$$ $$a + b = 64$$ Instead you have ...
3
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2answers
77 views

Do there exist shorter ways of solving for $x$ (or other smarter methods)?

Problem: If$$\frac 13\Bigr(\sqrt [3] {6\big(9+5\sqrt{3}\big)}+\sqrt [3] {6\big(9-5\sqrt{3}\big)}\Bigr)=2,\tag1$$ then $$\sqrt [3] {\frac x9}\bigr(\sqrt [3] {9+5\sqrt{3}}+\sqrt [3] {9-5\sqrt{3}}\bigr)=...
0
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3answers
73 views

How could I find the value of $x$ without squaring both sides given the equation $\sqrt{8} + \sqrt{18} = \sqrt{x}$? [closed]

Our class got a challenge question on a recent test, that no one has been able to figure out. We know $x = 50$, but everyone seems to get stuck when they simplify it to $5\sqrt{2} = \sqrt{x}$. Any ...
15
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7answers
2k views

Sum of cube roots of complex conjugates

When solving the following cubic equation: $$x^3 - 15x - 4 = 0$$ I got one of the solutions: $$x = \sqrt[3]{2 {\color{red}+} 11i} + \sqrt[3]{2 {\color{red}-} 11i}$$ When I calculated it with a ...
0
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8answers
217 views

Solving Radical Inequalities

I recently had an interesting discussion with my Algebra II teacher about solving the following inequality: $\sqrt{x}>-4$. As far as we could tell, squaring both sides results in $x>16$. However,...
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2answers
1k views

Difficult Radical Equation [closed]

I am stuck in the following radical equation $$ (4x-1)\sqrt{x^3+1}=2x^3+2x+1. $$ SOLUTION I have tried based on the guidance of the commenters and I think that the following solution is rather ...