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Questions tagged [radical-equations]

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

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Why do the solutions to $x^2 + 2x + 8\sqrt{x^2 + 2x + 21} - 41 = 0$ change when the equation is manipulated? [duplicate]

Starting with: $$x^2 + 2x + 8\sqrt{x^2 + 2x + 21} - 41 = 0 \tag{1}$$ If I try to simplify without substitution, by moving the root to the other side, squaring both sides, gathering like terms, I end ...
43Tesseracts's user avatar
1 vote
1 answer
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System of two polynomial equations in two unknowns

Solve for positive reals $(x,y)$ the two equations: $$ (17 y^2 - 13 x^2) (y-x) = 55\\ 3 y^2 - x^2 = 11 $$ One can first check the possible number of solutions. The second equation requires $y \ge \...
Andreas's user avatar
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Quotient of radical expressions has removable singularities

Similar to this post: Zero set of nested radicals, my question deals with functions on $\mathbb{R}$ that consist of nested radicals and polynomial functions. Is the following true? Let $P,Q$ be two ...
hbghlyj's user avatar
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Is the de Moivre's formula only intended to be used for unknown values of the input $z$, but not to fixed values of $z$?

This question is related to: What's the correct way of defining the use of square root symbol? As far as I know, the radical symbol $\sqrt{}$ only denotes the principal square root, even in the ...
Ronald Becerra's user avatar
3 votes
1 answer
62 views

Prove that there are infinitely many distinct natural numbers $a,b$ for which $\sqrt{a+b}, \sqrt{a-b}$ are simultaneously rational

the question Prove that there are infinitely many distinct natural numbers $a,b$ for which $\sqrt{a+b}, \sqrt{a-b}$ are simultaneously rational. the idea A radical is rational only if the number below ...
IONELA BUCIU's user avatar
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4 answers
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solution-verification | Show that if $a$ and $q$ are natural numbers and the number $(a+\sqrt{q})(a+\sqrt{q+1})$ is rational, then $q=0$.

the question Show that if $a$ and $q$ are natural numbers and the number $(a+\sqrt{q})(a+\sqrt{q+1})$ is rational, then $q=0$. the idea for the number to be rational both members have to be rational (*...
IONELA BUCIU's user avatar
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1 answer
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square root of x equals -1

I read that $\sqrt{x} = -1$ has no solution because after we square both sides we get $x = 1,$ which isn't a correct solution. But doesn't writing $-1$ as $i^2$ give the solution $x = i^4$ ? $$\sqrt{x}...
LukaMaths's user avatar
18 votes
6 answers
1k views

What are some obscure radical identities?

So there are several trigonometric identities, some very well know, such as $\cos(x) = 1 - 2\sin^2(\frac{x}{2})$ and some more obscure like $\tan(\frac{\theta}{2} + \frac{\pi}{4}) = \sec(\theta)+\tan(\...
karlabos's user avatar
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2 votes
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What is the rate of convergence of the following sequence (equation with a finite number of nested radicals)?

Let $f(x)=\sqrt{1-x^2}$, $b = 1/\sqrt{2}$. The sequence $(E_n)_{n=1}^{\infty}$ is defined as the solution to the following equation : $$f(E_n - f(E_n -f(E_n - ....-f(E_n - b)))) = E_n -1,$$ where the ...
Marc_Adrien's user avatar
1 vote
2 answers
126 views

Are there any simpler ways to determine the solution for $\sqrt{x+\sqrt{x}}=1$ without back substitution checks?

A weak condition by inspection: $x>0$. \begin{gather} \sqrt{x+\sqrt x} = 1\\ x+\sqrt x = 1\\ \sqrt x = 1-x\\ x = 1-2x+x^2\\ x^2 - 3x + 1 =0\\ x=\frac{3\pm\sqrt5}{2} \end{gather} As both satisfy ...
D G's user avatar
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2 votes
1 answer
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A radical equation $(2x+1)^{2/3}+(2x-1)^{2/3}-2x^{2/3}=2^{1/3}$

Solve the equation $(2x+1)^{2/3}+(2x-1)^{2/3}-2x^{2/3}=2^{1/3}$. I am looking for real roots. The graph of the equation tell us there are 4 solutions: roughly at $\pm0.09, \pm 1.64$, but I want to ...
Sean Ian's user avatar
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5 votes
4 answers
344 views

Any way to solve $\sqrt{x} + \sqrt{x+1} + \sqrt{x+2} = \sqrt{x+7}$?

I was solving a radical equation $x+ \sqrt{x(x+1)} + \sqrt{(x+1)(x+2)} + \sqrt{x(x+2)} = 2$. I deduced it to $\sqrt{x } + \sqrt{x+1} + \sqrt{x+2} = \sqrt{x+7}.$ Answer is $\frac1{24}$. The first ...
Utkarsh's user avatar
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1 vote
1 answer
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If every square root has positive and negative solutions, then is $-2 = 2\sqrt1$?

Since every square root has 2 possible solutions, one positive and one negative. Then wouldn't that happen every time you have a square root? Let's say for example: If $x + 1 = 2\sqrt{x+4}$ then $x$ ...
parpar8090's user avatar
1 vote
0 answers
130 views

Transforming a specific radical equation to a polynomial equation

I have this equation: $$0=\frac{8}{\sqrt{30^2-w^2}}+\frac{8}{\sqrt{20^2-w^2}}-1$$ But I need to express it as a polynomial equation, or an equivalent equation that is also polynomial, I have tried ...
paez49's user avatar
  • 11
4 votes
2 answers
160 views

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 ...
Yaroslav Nikitenko's user avatar
2 votes
1 answer
141 views

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 ...
Darryl Noakes's user avatar
1 vote
3 answers
169 views

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{...
Hypergeometricx's user avatar
0 votes
3 answers
458 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 ...
Mr. Engineer's user avatar
0 votes
1 answer
111 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}{...
user avatar
4 votes
2 answers
492 views

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)=...
Dick Grayson's user avatar
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3 votes
5 answers
347 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 ...
user avatar
0 votes
0 answers
55 views

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 ...
bingus's user avatar
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1 vote
2 answers
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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 ...
Parham Moieni's user avatar
5 votes
3 answers
824 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 $...
Chris Christopherson's user avatar
7 votes
4 answers
282 views

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-...
user avatar
5 votes
4 answers
331 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}=-...
ShootinLemons's user avatar
3 votes
4 answers
111 views

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 $3$ $$(...
nickh's user avatar
  • 475
4 votes
2 answers
194 views

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 ...
zeraoulia rafik's user avatar
1 vote
0 answers
27 views

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 ...
mathable's user avatar
  • 444
2 votes
2 answers
178 views

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^...
MethStudent's user avatar
1 vote
3 answers
713 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 ...
user avatar
0 votes
3 answers
132 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}-...
Max Villafranca's user avatar
2 votes
8 answers
821 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 ...
Math Student's user avatar
  • 5,352
1 vote
4 answers
164 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[\...
Math Student's user avatar
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2 votes
4 answers
144 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 ...
Math Student's user avatar
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0 votes
3 answers
108 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 ...
Math Student's user avatar
  • 5,352
0 votes
1 answer
73 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{...
Anthony's user avatar
  • 3,758
1 vote
1 answer
109 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 ...
bvggy's user avatar
  • 21
0 votes
2 answers
53 views

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{...
User154's user avatar
  • 361
2 votes
2 answers
147 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} &...
User154's user avatar
  • 361
6 votes
1 answer
2k 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 ...
Hugo Faurand's user avatar
4 votes
3 answers
151 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 ...
user avatar
0 votes
1 answer
42 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 ...
Vinicius L. Beserra's user avatar
1 vote
1 answer
141 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 ...
user avatar
5 votes
5 answers
301 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 ...
mku's user avatar
  • 107
3 votes
2 answers
140 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-...
Oliv's user avatar
  • 33
-1 votes
1 answer
108 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 ...
user703237's user avatar
2 votes
1 answer
90 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 ...
MR_Q's user avatar
  • 158
1 vote
1 answer
240 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}}...
Plinth's user avatar
  • 143
0 votes
1 answer
222 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 ...
ahmed allam's user avatar