For equations in which the variable(s) is/are under a radical.
Questions tagged [radical-equations]
93 questions
19
votes
6 answers
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 $\tan(\frac{\theta}{2} + \frac{\pi}{4}) = \sec(\theta)+\tan(\theta)$.
You can also find some logarithmic…
karlabos
- 1,359
16
votes
7 answers
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 hand calculator, it turned out to be exactly $4$. And…
BarbaraKwarc
- 466
13
votes
6 answers
solving $\sqrt{3-\sqrt{3+x}}=x$.
Can we solve the following equation in $\mathbb{R}$ without expanding it into a fourth degree equation :
$$ \sqrt{3-\sqrt{3+x}} = x.$$
squaring both sides and squaring again is the only thing I could done, If you have any other idea just post…
user55386
7
votes
4 answers
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, $$(y-2)(3y^{14}-(y-2)^{15})=0$$
One solution is $y = 2$ and another…
user979855
6
votes
5 answers
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 equation has two solutions however the latter one has…
Utkarsh
- 1,966
6
votes
1 answer
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 equation by squaring and using $(a-b)(a+b)=a^2…
Hugo Faurand
- 157
6
votes
2 answers
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 traditionaly, I get a polynomial with exponent to the 4th…
Karlovsky120
- 213
- 1
- 9
5
votes
3 answers
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 $b$. Then what allows us to say the following?…
Chris Christopherson
- 1,318
- 11
- 27
5
votes
4 answers
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 $\sqrt{2x+3}=-1$. So does $x=-1$ technically count as a solution or…
ShootinLemons
- 287
5
votes
5 answers
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 need someone to direct me.
I was thinking if the…
mku
- 117
5
votes
1 answer
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 appreciated. Thanks
Nguyen Thy
- 613
5
votes
3 answers
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 ideas of how should this be solved?
I got…
Aleex_
- 101
5
votes
2 answers
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
$\displaystyle n!\approx \bigg(\frac{n}{e}\bigg)^n\sqrt{2\pi n}$ for laege…
DXT
- 12,047
5
votes
1 answer
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 simplify them out, but it just became more…
4
votes
2 answers
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 be functions of $x$, and there could be more…
