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 ...
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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 \...
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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 ...
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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 ...
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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 ...
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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 (*...
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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,$$
where the ...
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Are there any simpler ways to determine the solution for $\sqrt{x+\sqrt{x}}=1$ without back substitution checks?
A weak condition by inspection: $x>0$.
\begin{gather}
\sqrt{x+\sqrt x} = 1\\
x+\sqrt x = 1\\
\sqrt x = 1-x\\
x = 1-2x+x^2\\
x^2 - 3x + 1 =0\\
x=\frac{3\pm\sqrt5}{2}
\end{gather}
As both satisfy ...
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A radical equation $(2x+1)^{2/3}+(2x-1)^{2/3}-2x^{2/3}=2^{1/3}$
Solve the equation $(2x+1)^{2/3}+(2x-1)^{2/3}-2x^{2/3}=2^{1/3}$.
I am looking for real roots. The graph of the equation tell us there are 4 solutions: roughly at $\pm0.09, \pm 1.64$, but I want to ...
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Any way to solve $\sqrt{x} + \sqrt{x+1} + \sqrt{x+2} = \sqrt{x+7}$?
I was solving a radical equation $x+ \sqrt{x(x+1)} + \sqrt{(x+1)(x+2)} + \sqrt{x(x+2)} = 2$. I deduced it to $\sqrt{x } + \sqrt{x+1} + \sqrt{x+2} = \sqrt{x+7}.$
Answer is $\frac1{24}$.
The first ...
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If every square root has positive and negative solutions, then is $-2 = 2\sqrt1$?
Since every square root has 2 possible solutions, one positive and one negative. Then wouldn't that happen every time you have a square root?
Let's say for example: If $x + 1 = 2\sqrt{x+4}$ then $x$ ...
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Transforming a specific radical equation to a polynomial equation
I have this equation:
$$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 ...
