Questions tagged [radical-equations]
For equations in which the variable(s) is/are under a radical.
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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 $x = i^4$ ?
$$\sqrt{x}...
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What are some obscure radical identities?
So there are several trigonometric identities, some very well know, such as $\cos(x) = 1 - 2\sin^2(\frac{x}{2})$ and some more obscure like $\tan(\frac{\theta}{2} + \frac{\pi}{4}) = \sec(\theta)+\tan(\...
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What is the rate of convergence of the following sequence (equation with a finite number of nested radicals)?
Let $f(x)=\sqrt{1-x^2}$, $b = 1/\sqrt{2}$. The sequence $(E_n)_{n=1}^{\infty}$ is defined as the solution to the following equation :
$$f(E_n - f(E_n -f(E_n - ....-f(E_n - b)))) = E_n -1,$$
where the ...
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Are there any simpler ways to determine the solution for $\sqrt{x+\sqrt{x}}=1$ without back substitution checks?
A weak condition by inspection: $x>0$.
\begin{gather}
\sqrt{x+\sqrt x} = 1\\
x+\sqrt x = 1\\
\sqrt x = 1-x\\
x = 1-2x+x^2\\
x^2 - 3x + 1 =0\\
x=\frac{3\pm\sqrt5}{2}
\end{gather}
As both satisfy ...
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How to find the roots to this trigonometric equation
I need help with this equation:
$$\frac{5}{4}\cdot\left(150\text{cos}(x)-40\text{sin}(x)\right)=150\cdot\frac{\left(10-5\sqrt{3}\ \right)}{4}$$
WolframAlpha gave me exact values but I need to know the ...
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A radical equation $(2x+1)^{2/3}+(2x-1)^{2/3}-2x^{2/3}=2^{1/3}$
Solve the equation $(2x+1)^{2/3}+(2x-1)^{2/3}-2x^{2/3}=2^{1/3}$.
I am looking for real roots. The graph of the equation tell us there are 4 solutions: roughly at $\pm0.09, \pm 1.64$, but I want to ...
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Any way to solve $\sqrt{x} + \sqrt{x+1} + \sqrt{x+2} = \sqrt{x+7}$?
I was solving a radical equation $x+ \sqrt{x(x+1)} + \sqrt{(x+1)(x+2)} + \sqrt{x(x+2)} = 2$. I deduced it to $\sqrt{x } + \sqrt{x+1} + \sqrt{x+2} = \sqrt{x+7}.$
Answer is $\frac1{24}$.
The first ...
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If every square root has positive and negative solutions, then is $-2 = 2\sqrt1$?
Since every square root has 2 possible solutions, one positive and one negative. Then wouldn't that happen every time you have a square root?
Let's say for example: If $x + 1 = 2\sqrt{x+4}$ then $x$ ...
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Transforming a specific radical equation to a polynomial equation
I have this equation:
$$0=\frac{8}{\sqrt{30^2-w^2}}+\frac{8}{\sqrt{20^2-w^2}}-1$$
But I need to express it as a polynomial equation, or an equivalent equation that is also polynomial, I have tried ...
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free software for radical algebraic equations
I want to study an algebraic curve defined by equations of the form
$$ a_1 \sqrt{f_1(x)} + ... + a_n \sqrt{f_n(x)} = 0, $$
where $x$ is a real variable and $f_i$ are polynomials. $ a_1,... a_n $ could ...
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How to interpret an expression when the radical doesn't extend over anything?
I have a school assignment which includes solving this problem from a scanned document:
Equivalent:
Given that $m = { \sqrt{} l - n^2 \over n }$,
express $n$ in terms of $m$.
How do interpret this ...
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A golden question $\sqrt{2+\sqrt{2-x}}=\sqrt{x-\frac 1x} + \sqrt{1-\frac 1x}$
How would you solve this problem for real $x$?
$$\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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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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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
$$\to \frac{x^2 +1-1}{...
user983440
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Understanding Cardano's Formula
In deriving his formula, Cardano arrives at the equation $y^3+py+q=0$. By substituting $y=\sqrt[3]{u}+\sqrt[3]{v}$, he gets the equation $(u+v+q)+(\sqrt[3]{u}\sqrt[3]{v})(3\sqrt[3]{u} \sqrt[3]{v} +p)=...
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Solving $(x+1)\sqrt{2(x^2 + 1)} + \sqrt{6x^2 + 18x +12} = \frac{3x^2 + 7x + 10}{2}$
Today, I came across this problem.
$$(x+1)\sqrt{2(x^2 + 1)} + \sqrt{6x^2 + 18x +12} = \dfrac{3x^2 + 7x + 10}{2}$$
We are asked to find the possible values of $x$ satisfying this equation.
The first ...
user983440
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Expressing the solution to a rational expression in radical form
I'm trying to find the solution to this equation
$$
-\frac{3}{r} + \frac{8}{r^3} = \frac{\sqrt{2}-1}{2}
$$
but I haven't been able to find a solution in radical form. Although I've found the solutions ...
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Why can't a negative base be raised to a non-integer exponent?
Why can't we do this?
Example: $(-1) ^ {1/3}$
Math definitions are based on a definite logic. What is the logic here?
Can you give me some examples where it violates the equation?
I'm just a high ...
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If $\sqrt{a}\sqrt{b}=\sqrt{ab}$ only holds for positive real $a$ & $b$, then why can we say $\sqrt{-a}=\sqrt{-1\cdot a}=\sqrt{-1}\sqrt{a}=i\sqrt{a}$?
I am a little bit bummed that I have this question as I'm sure it has been asked before (I couldn't find the answer) but...
If $\sqrt{a}\sqrt{b} = \sqrt{ab}$ is only true for positive reals $a$ and $...
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Solve for $x$ in $\sqrt{x+2\sqrt{x+2\sqrt{x+2\sqrt{3x}}}} = x$
Solve for $x$:
$$\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-...
user979855
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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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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 $3$
$$(...
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Pairs of integers $ (x,m)$ for which $\sqrt[3]{\sqrt[3]{x-2}+m}+\sqrt[3]{-\sqrt[3]{x-2}+m}=2$ hold?
Find all pairs of integers $(x,m)$ for which $$\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, $$ I come ...
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Solvable elements of a field extension
Suppose $K$ over $F$ is a field extension, and $\alpha \in K$.
My instructor says that "$\alpha$ is solvable over $F$ if there exists a radical extension $L$ of $F$ containing $\alpha$".
My ...
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Calculate the sum of all irrational roots of $4\sqrt[3]{8x- 3}- 8x^{3}- 3= 0$
Calculate the sum of all irrational roots of
$$4\sqrt[3]{8x- 3}- 8x^{3}- 3= 0$$
I'm not even sure how to begin here, I tried raising it to the power of three, tried writing $8x^{3}+ 3$ with $x^{3}+ y^...
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The set of all $x$ satisfying, $\sqrt{4x+1} + \sqrt{7-x} = 6 $, consists of:
The set of all $x$ satisfying, $\sqrt{4x+1} + \sqrt{7-x} = 6 $, consists of:
$A)$ Two rational numbers. $B)$ An irrational number. $C)$ Complex number. $D)$ None.
How to solve the above question ...
user880107
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How many zeros does a radical equation (eg, $X^{4/3}-5X^{2/3}+6=0$) have?
I want to know if there is a general rule that will give me the answer.
I'm not talking about crazy expressions under a radical, just simple
variables raised to a fractional exponent like:
$$X^{4/3}-...
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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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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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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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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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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:
$$
\frac{1}{X}\bigg\lfloor-\frac{3}{2}+\frac{1}{2}\sqrt{8X+9}\bigg\rfloor=\frac{1}{N} \label{a}\tag{...
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Can fractional/decimal radicals/roots exist?
For questions like "What is the 1/2th root of x would the answer be $x^2$?
My logic is that since $$
\sqrt[\cfrac{1}{2}]{x}=x^{1/{(\cfrac{1}{2}})}
$$
Which simplifies to $x^2$.
So as a general ...
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Equivalence of two radical equations without certain conditions - correctness of method
I have question related to the following example: $\sqrt{22-x} - \sqrt{10-x}=2$.
First question:
Do I first need conditions $22-x \geq 0 $ and $ 10-x \geq 0$ to obtain the equivalent equation: $\sqrt{...
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Why substitution in irrational equation doesn't give equivalent equation?
I have two examples of irrational equations:
The first example: $\sqrt[3]{3-x} + \sqrt[3]{6+x}=3$
In solution, they take cube of both sides and do following:
\begin{eqnarray*}
&\sqrt[3]{3-x} &...
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Solving $\sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$
I have to solve this irrational equation on $\mathbb{R}$ :
$$ \sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$$
I tried to do a substitution with $u=1-x$ but the only things I manage to reach is the following ...
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Fast way to solve $4 = \sqrt[3] {x+10}-\sqrt[3] {x-10}$
The question is this:
$4 = \sqrt[3] {x+10}-\sqrt[3] {x-10}$
For some reason, I keep on getting 289/3, even though it is the wrong answer. This is from a timed test, and my way is wrong and extremely ...
user807252
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Find the roots for y
$$-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}$$
This is from a timed competition, and I would like to know the fastest way to do it. I'm not sure, but is ...
user807252
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Semicircle Question
I need help with the question in the image. I just need someone to help by pointing me in the right direction. I don't want a full solution. I want to try to work out this question myself but I just ...
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Solving $\sqrt[3]{x+1} - \sqrt[3]{x-1} = \sqrt[3]{x^2-1}$ for real $x$
Solve the equation in the Real number system:
$$\sqrt[3]{x+1} - \sqrt[3]{x-1} = \sqrt[3]{x^2-1}$$
I have attempted using $(A-B)^3 = A^3 - B^3 - 3.A.B.(A-B)$ with $A = \sqrt[3]{x+1}$ , $B = \sqrt[3]{x-...
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Integrate $\frac{x}{(x^4+1)^2 \sqrt{(x^2-x+1)}}$. [closed]
Evaluate the indefinite integral, $$\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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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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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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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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$K=\mathbb{Q}(\sqrt[8]{2},i)$ 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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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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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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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 ...
