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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### 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 $$\sqrt{45x^2-30x+1}=7+6x-9x^2.$$ 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 $x\in\left[\... • 5,352 2 votes 4 answers 144 views ### Solve the equation$\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 ... • 5,352 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 ... • 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: $$\frac{1}{X}\bigg\lfloor-\frac{3}{2}+\frac{1}{2}\sqrt{8X+9}\bigg\rfloor=\frac{1}{N} \label{a}\tag{... • 3,758 1 vote 1 answer 109 views ### Can fractional/decimal radicals/roots exist? For questions like "What is the 1/2th root of x would the answer be x^2? My logic is that since$$ \sqrt[\cfrac{1}{2}]{x}=x^{1/{(\cfrac{1}{2}})} $$Which simplifies to x^2. So as a general ... • 21 0 votes 2 answers 53 views ### Equivalence of two radical equations without certain conditions - correctness of method I have question related to the following example: \sqrt{22-x} - \sqrt{10-x}=2. First question: Do I first need conditions 22-x \geq 0 and 10-x \geq 0 to obtain the equivalent equation: \sqrt{... • 361 2 votes 2 answers 147 views ### Why substitution in irrational equation doesn't give equivalent equation? I have two examples of irrational equations: The first example: \sqrt[3]{3-x} + \sqrt[3]{6+x}=3 In solution, they take cube of both sides and do following: \begin{eqnarray*} &\sqrt[3]{3-x} &... • 361 6 votes 1 answer 2k views ### Solving \sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2} I have to solve this irrational equation on \mathbb{R} :$$ \sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$$I tried to do a substitution with u=1-x but the only things I manage to reach is the following ... • 157 4 votes 3 answers 151 views ### Fast way to solve 4 = \sqrt[3] {x+10}-\sqrt[3] {x-10} The question is this: 4 = \sqrt[3] {x+10}-\sqrt[3] {x-10} For some reason, I keep on getting 289/3, even though it is the wrong answer. This is from a timed test, and my way is wrong and extremely ... 0 votes 1 answer 42 views ### Find the roots for y$$-1=(0.55)\cdot[1+(y+1)^2]^{\frac{3}{2}}$$I got stuck with this expression. I have l some difficulty in leanding with some algebraic manipulation. What should I do to solve this equation?? I tried ... 1 vote 1 answer 141 views ### simplify the equation \frac{36}{\sqrt{x}} + \frac{9}{\sqrt{y}} = 42-9\sqrt{x}-\sqrt{y} This is the question:$$\frac{36}{\sqrt{x}} + \frac{9}{\sqrt{y}} = 42-9\sqrt{x}-\sqrt{y}$$This is from a timed competition, and I would like to know the fastest way to do it. I'm not sure, but is ... 5 votes 5 answers 301 views ### Semicircle Question I need help with the question in the image. I just need someone to help by pointing me in the right direction. I don't want a full solution. I want to try to work out this question myself but I just ... • 107 3 votes 2 answers 140 views ### Solving \sqrt[3]{x+1} - \sqrt[3]{x-1} = \sqrt[3]{x^2-1} for real x Solve the equation in the Real number system:$$\sqrt[3]{x+1} - \sqrt[3]{x-1} = \sqrt[3]{x^2-1}$$I have attempted using (A-B)^3 = A^3 - B^3 - 3.A.B.(A-B) with A = \sqrt[3]{x+1} , B = \sqrt[3]{x-... • 33 -1 votes 1 answer 108 views ### Integrate \frac{x}{(x^4+1)^2 \sqrt{(x^2-x+1)}}. [closed] Evaluate the indefinite integral,$$\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 ... 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 ... • 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}}...
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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 ...