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Questions tagged [radical-equations]

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0
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1answer
52 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< \...
0
votes
0answers
16 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)...
0
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2answers
37 views

Confusing regarding a nested radical equation

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 that this equation is ...
2
votes
1answer
55 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. ...
2
votes
3answers
173 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 \...
0
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1answer
38 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_{...
1
vote
2answers
48 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?
-1
votes
3answers
44 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
1
vote
4answers
91 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 ...
0
votes
3answers
28 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
votes
3answers
52 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
votes
3answers
199 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 ...
3
votes
2answers
107 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 ...
-2
votes
3answers
87 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
votes
2answers
68 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}}...
0
votes
3answers
71 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 ...
14
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7answers
1k 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
108 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,...
-2
votes
2answers
684 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 ...
2
votes
3answers
65 views

$\sqrt{x}+\sqrt{2-x}\geq a$ has a solution with interval $\leq 3$. [Solve for “a”]. [closed]

I need to solve this inequality. The first part of the problem was to solve for $a=3$ implies $x\in [0,2]$. This makes me think that I will somehow use this. I try to find the solution for the second ...
2
votes
2answers
402 views

Elementary method to solve this equation

I have to describe ''the sign of the root(s)''$$\sqrt[3]{3+\sqrt{x}}+\sqrt[3]{3-x}-\sqrt[3]{6}=0$$ for k-11 students , who did not learned derivation. I can solve the equation by taking $f(x)=\sqrt[3]...
3
votes
1answer
406 views

Solve the radical equation $ x\sqrt{x^2+5} + (2x+1)\sqrt{4x^2+4x+6}=0.$

Solve the following equation: $$ x\sqrt{x^2+5} + (2x+1)\sqrt{4x^2+4x+6}=0.$$ I wanted to solve this equation. First I tried to change the equations under the roots to the complete square to ...
4
votes
4answers
996 views

How to find the solution of $\sqrt{x^2+12x+35} \geq x-10$?

I want to ask about radical inequality problem. Here's the question: Find the solution sets for $\sqrt{x^2+12x+35}\geq x-10$ My attempts to tackle this problem is like this: Firstly I try to ...
2
votes
1answer
66 views

Need help solving the following equation

I'm having difficulty solving the following equation for $x$: $$\sqrt[m]{(1+x)^2}-\sqrt[m]{(1-x)^2}=\sqrt[m]{1-x^2}$$ I have tried a few substitutions but that didn't seem to get me anywhere. Can ...
3
votes
1answer
118 views

“Extraneous solution” solves original equation

For the given equation: $$x - 10 = \sqrt{9x}$$ when one simplifies, through the following steps: \begin{align*} x^2 - 20x + 100 &= 9x\\ x^2 - 29x + 100 &= 0\\ x &= 25, 4 \end{align*} ...
2
votes
2answers
113 views

Surds question grade 10

I am a student and need help answering this question. I need a step by step solution. Simplify: $ \sqrt {18}$ - $ \sqrt {9}$ What I tried: ($ \sqrt {9}$ × $ \sqrt {2}$) - 9 = (3$ \sqrt {2}$ ) - ...
0
votes
2answers
464 views

Inequality with square root on both sides

I have equation like this: $$\sqrt{\vphantom{|}\ 3x^2-7x-20}<\sqrt{\vphantom{|}\ 8x+22}$$ I'm unsure how to solve it. I'm guessing I have to square both sides, but I don't know what happens with ...
6
votes
2answers
587 views

Trick to this square root equations

Okay, so this is a high school level assignment: $$ \sqrt{x+14}-\sqrt{x+5}=\sqrt{x-2}-\sqrt{x-7} $$ Here's a similar one: $$ \sqrt{x}+\sqrt{x-5}=\sqrt{x+7}+\sqrt{x-8} $$ When solving these ...